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Birmingham Small Arms Company

The Birmingham Small Arms Company Limited (BSA) was a major British industrial combine, a group of businesses manufacturing military and sporting firearms; bicycles; motorcycles; cars; buses and bodies; steel; iron castings; hand, power, and machine tools; coal cleaning and handling plants; sintered metals; and hard chrome process.

After the Second World War, BSA did not manage its business well, and a government-organised rescue operation in 1973 led to a takeover of such operations as it still owned. Those few that survived this process disappeared into the ownership of other businesses.

Machine-made guns
BSA began in June 1861 in the Gun Quarter, Birmingham, England. It was formed by a group of fourteen gunsmith members of the Birmingham Small Arms Trade Association specifically to manufacture guns by machinery. They were encouraged to do this by the War Office which gave the BSA gunsmiths free access to technical drawings and to the War Office's Board of Ordnance's Royal Small Arms Factory at Enfield. New machinery developed in the USA installed at Enfield had greatly increased its output without needing more skilled craftsmen. This new machinery brought to Birmingham the principle of the interchangeability of parts. BSA bought 25 acre of land at Small Heath, Birmingham, built a factory there and made a road on the site calling it Armoury Road. Their enterprise was rewarded in 1863 with an order for 20,000 Turkish infantry rifles.

The system of management of BSA was changed in 1863 when shareholders elected a Board of Directors: Joseph Wilson, Samuel Buckley, Isaac Hollis, Charles Playfair, Charles Pryse, Birmingham mayor Sir John Ratcliffe (c.1798-1864), Edward Gem, and J.F. Swinburn under the chairmanship of John Dent Goodman (1816-1900).

Erratic demand
The first War Office contract was not agreed until 1868. In 1879, without work, the factory was shut for a year.

Bicycles
The next year, 1880, BSA branched out into bicycle manufacture. The gun factory proved remarkably adaptable to the manufacture of cycle parts. What cycles needed was large quantities of standard parts accurately machined at low prices. In 1880, BSA manufactured the. In the 1880s, the company began to manufacture safety bicycles on their own account and not until 1905 was the company's first experimental motorcycle constructed. Bicycle production ceased in 1887 as the company concentrated on producing the Lee–Metford magazine-loading rifle for the War Office which was re-equipping the British Army with it. The order was for 1,200 rifles per week. BSA recommenced manufacturing bicycles on their own behalf from 1908. BSA Cycles Ltd was set up in 1919 for the manufacture of both bicycles and motorcycles. BSA sold the bicycle business to Raleigh in 1957 after separating the bicycle and motorcycle business in 1953.

Bicycle components
In 1893, BSA commenced making bicycle hubs and continued to supply the cycle trade with bicycle parts up to 1936. BSA bought The Eadie Manufacturing Company of Redditch in 1907 and so began to manufacture the Eadie two speed hub gear and the Eadie coaster brake hub. BSA also signed an agreement with the Three Speed Gear Syndicate in 1907 to manufacture a 3 speed hub under licence. This was later classified as the Sturmey Archer Type X. BSA introduced a 'Duo' hub in the late 1930s which was capable of one fixed gear and one gear with a freewheel. All BSA hub gear production temporarily ceased in 1939, until they recommenced making their 3 speed hub around 1945. The Eadie coaster hub made a brief return in 1953 on two BSA bicycle models. BSA forever ceased production of their hub gears in 1955.

Ammunition
BSA sold its ammunition business in 1897 to Birmingham Metal and Munitions Company Limited part of the Nobel-Dynamite Trust, through Kynoch a forerunner of ICI.

Sparkbrook Royal Small Arms Factory
In 1906, Frank Dudley Docker was appointed a director of the company. By the autumn of that year BSA was in some difficulty. They had purchased the Sparkbrook Royal Small Arms Factory from the War Office, and in return, the War Office undertook to give BSA a quarter of all orders for Lee–Enfield rifles. But, the War Office did not honour their undertaking. The ensuing financial crisis did not prevent BSA from signing an agreement to purchase control of bicycle component manufacturer, the Eadie Manufacturing Company of Redditch, on 11 February 1907. That decision was ratified by the shareholders of both companies at separate Extraordinary General Meetings held in the Grand Hotel, Birmingham on 27 February 1907. Albert Eadie became a BSA director, a post he held until his death in 1931.

Sporting firearms
The very variable military market was now supported by sales of target military rifles, sporting rifles, various patterns of miniature rifles and air rifles. Aperture sights were in demand for Bisley and other military rifle meetings.

Motorcycles
Motor bicycles were added to bicycle products in 1910. The BSA $3 1/2$ hp was exhibited at the 1910 Olympia Show, London for the 1911 season. The entire BSA production sold out in 1911, 1912 and 1913.

BSA cars
In an effort to make use of the Sparkbrook factory BSA established a motorcar department there. An independent part of it was occupied by Lanchester Motor Company. The first prototype automobile was produced in 1907. The following year, marketed under BSA Cycles Ltd, the company sold 150 automobiles and again began producing complete bicycles on its own account. By 1909, it was clear the new motorcar department was unsuccessful; an investigation committee reported to the BSA Board on the many failures of its management and their poor organisation of production.

Daimler cars and trucks
Dudley Docker had joined the board in 1906 and was appointed deputy chairman of BSA in 1909. He had made a spectacular financial success of a merger of five large rolling-stock companies in 1902 and become the leader of the period's merger movement. Believing he could buy the missing management skills that could not be found within BSA, he started merger talks with The Daimler Company Limited of Coventry. Daimler and Rover were then the largest British car producers. Daimler was immensely profitable. After its capital reconstruction in 1904, Daimler's profits were 57 per cent and 150 per cent returns on invested capital in 1905 and 1906. The attraction for Daimler shareholders was the apparent stability of BSA.

So in 1910, BSA purchased Daimler with BSA shares but Docker who negotiated the arrangements either ignored or failed in his assessment of their consequences for the new combine. The combine was never adequately balanced or co-ordinated. One of the financial provisions obliged Daimler to pay BSA an annual dividend of £100,000 representing approximately 40 per cent of the actual cash BSA had put into Daimler. This financial burden deprived Daimler of badly needed cash to fund development, forcing the Daimler company to borrow money from the Midland Bank.

BSA had still not recovered financially from the earlier purchase of Royal Small Arms factory at Sparkbrook and BSA were not in a position to finance Daimler, nor had either company ample liquid resources. BSA went ahead with motorcycle production in 1910, their first model available for the 1911 season. In 1913, the BSA group were compelled through pressure from the Midland Bank to make a capital issue of 300,000 preference shares. In the short term this was to solve the liquidity issue but further diluted the group's capitalisation.

Dudley Docker retired as a BSA director in 1912 and installed Lincoln Chandler on the BSA board as his replacement. Dudley Docker liked to draw a comparison between the BSA–Daimler merger he engineered and that of his 1902 merger of Metropolitan Carriage Wagon & Finance Company and Patent Shaft. However, there was not the integration of facilities in the BSA–Daimler case, nor was there a reorganisation of either BSA or Daimler and in view of the earlier criticism contained in the 1909 report of the investigation committee, BSA continued to produce cars of their own using Daimler engines. In 1913, Daimler employed 5,000 workers to manufacture 1,000 vehicles, an indication that things were not well.

Steel bodies
In 1912, BSA would be one of two automobile manufacturers pioneering the use of all-steel bodies, joining Hupmobile in the US.

Lewis Automatic Machine Gun
In 1913, BSA undertook to manufacture quick-firing machine guns for the Lewis Automatic Arms Company whose rights covered the world except for the American Continent.

First World War
During the First World War, the company returned to arms manufacture and greatly expanded its operations. BSA produced rifles, Lewis guns, shells, bicycles, motorcycles and, through Daimler, aero engines, aircraft and other vehicles for the war effort as well as machine tools.

Following the Armistice the BSA group was described by its chairman as:
 * Three distinct legal entities: BSA guns, BSA (machine) tools and BSA cycles all at Sparkbrook, Smallheath and Redditch
 * Daimler at Coventry
 * William Jessop and Savilles the two steel-making companies in Sheffield
 * Daimler Hire and Burton Griffiths in London

Motorcycles

In November 1919, BSA launched their first 50-degree vee-twin, Model E, 770cc side valve (6-7 hp) motorcycle for the 1920 season. The machine had interchangeable valves, total loss oil system with mechanical pump and an emergency hand one. Retail price was £130. Other features were Amac carburettor, chain drive, choice of magneto or Magdyno, 7-plate clutch, 3 speed gear box with kickstarter and new type of cantilever fork

Aviation
During the war Daimler had built enormous numbers of aero engines and aircraft and by the end was building 80 Airco de Havilland bombers a month. In February 1920 BSA amalgamated with what was the world's largest aircraft manufacturer, Aircraft Manufacturing Company (Airco), Airco's main plant at Hendon had employed between 7,000 and 8,000 people. The Airco group of companies had turned out a new aircraft every 45 minutes.

Within days BSA discovered Airco was in a far more serious financial state than George Holt Thomas had revealed. Holt Thomas was immediately dropped from his new seat on the BSA board and all BSA's new acquisitions were placed in the hands of a liquidator. Some of the businesses were allowed to continue for some years, Aircraft Transport and Travel's assets being eventually rolled into Daimler Air Hire to make Daimler Airway Limited. BSA failed to pay a dividend for the following four years while it tried to recover from its losses. Some relief was achieved when in March 1924 Daimler Airway and its management became the major constituent of Imperial Airways. As well as the Daimler car range, BSA Cycles Ltd re-entered the car market under the BSA name in 1921 with a V-twin engined light car followed by four-cylinder models up to 1926, when the name was temporarily dropped. In 1929 a new range of 3- and 4-wheel cars appeared and production of these continued until 1936.

By the end of 1924 difficult economic conditions left the bulk of BSA profits coming from cars and cycles. There were no sales of arms for military purposes in spite of large new facilities built at Government's request. The shares in Pennsylvania's Jessop Steel Co were disposed of without loss. During 1928 there was a drastic reorganisation of the business of some BSA subsidiaries.

By 1930 the BSA Group's primary activities were BSA cycles and Daimler vehicles.

Car production under the BSA name ceased in the 1930s. BSA remained the largest manufacturer of motorcycles but the market was less than half the size of the late 1920s and production was unprofitable yet the value of BSA's motor cars and cycles was now more than half group turnover.

Lanchester
In 1931 the Lanchester Motor Company at Sparkbrook was acquired  and production of their cars transferred to Daimler's Coventry works. The first new product was a version of the Daimler Light Twenty or 16/20 and called Lanchester 15/18.

Economic conditions began to improve in the mid 1930s and BSA's activities and profits all grew sharply. International tensions added more activity. An aero engine shadow factory was built and entered production during 1938. Motor cycle sales shrank but BSA maintained its relative position. 1937's new British registrations fell from 57,000 to 46,500. Defence and military equipment including Daimler's Scout car were in heavy demand in Britain and in export markets.

Armaments
In the 1930s, the board of directors authorised expenditure on bringing their arms-making equipment back to use – it had been stored at company expense since the end of the Great War in the belief that BSA might again be called upon to perform its patriotic duty. In 1939, BSA acquired the blueprints for a submachine gun designed by Hungarian arms designer Pál Király and the rights to manufacture it. Examples were produced in 9mm Mauser Export calibre. It was estimated that it would only cost £5 each to manufacture: by comparison, the Thompson submachine guns bought after the start of hostilities cost around £50, while SMLE rifles and the later Sten submachine guns cost £7. 15s and £2. 10s respectively in 1943. However the trials did not lead to acceptance; referring to the complex trigger mechanism, Frank Hobart said "no soldier could have coped with this watchmaker's dream",

Second World War
By the outbreak of the Second World War, BSA Guns Ltd at Small Heath, was the only factory producing rifles in the UK. The Royal Ordnance Factories did not begin production until 1941. BSA Guns Ltd was also producing .303 Browning machine guns for the Air Ministry (the main aircraft armament at the time) at the rate of 600 guns per week in March 1939 and Browning production was to peak at 16,390 per month by March 1942. The armed forces had chosen the 500 cc side-valve BSA M20 motorcycle as their preferred machine. On the outbreak of war the Government requisitioned the 690 machines BSA had in stock as well as placing an order for another 8,000 machines. South Africa, Ireland, India, Sweden and the Netherlands also wanted machines.

The Government passed the Emergency Powers (Defence) Act 1939 on 24 August allowing the drafting of defence regulations affecting food, travel, requisitioning of land and supplies, manpower and agricultural production. A second Emergency Powers (Defence) Act was passed on 22 May 1940 allowing the conscription of labour. The fall of France had not been anticipated in Government planning and the encirclement of a large part of the British Expeditionary Force into the Dunkirk pocket resulted in a hasty evacuation of that part of the B.E.F following the abandonment of their equipment. The parlous state of affairs "no arms, no transport, no equipment" in the face of the threat of imminent invasion of Britain by Nazi forces was recorded by the Chief of the Imperial General Staff Field Marshal Sir Alan Brooke, 1st Viscount Alanbrooke in his diary entries of the 1/2 July 1940. The creation of the Home Guard (initially as the Local Defence Volunteers) following Anthony Eden's broadcast appeal to the Nation on Tuesday 14 May 1940 also created further demand for arms production to equip this new force. BSA, as the only rifle producer in Britain, had to step up to the mark and the workforce voluntarily went onto a seven-day week. Motorcycle production was also stepped up from 500 to 1,000 machines per week which meant a finished machine coming off the production line every 5 minutes. The motorcycle department had been left intact in 1939 due to demand which was doubled following Dunkirk. At the same time BSA staff were providing lectures and demonstrations on motorcycle riding and maintenance to 250,000 officers and men in all parts of the UK.

The BSA factory at Small Heath was bombed by the Luftwaffe on 26 August 1940 resulting in one high explosive bomb and a shower of incendiaries hitting the main barrel mill which was the only one operating on service rifles in the country, causing the unaffordable loss of 750 machine tools but fortunately no loss of life. Two further Blitz air raids took place on 19 and 22 November 1940. The air raid of 19 November did the most damage, causing loss of production and trapping hundreds of workers. Since BSA was the sole producer of the main aircraft armament, the resulting delays in productions reportedly caused most worries to PM Churchill among all the industrial damage during the Blitz.

Two BSA night-shift electricians, Alf Stevens and Alf Goodwin, helped rescue their fellow workers. Alf Stevens was awarded the George Medal for his selfless acts of bravery in the rescue and Alf Goodwin was awarded the British Empire Medal. Workers involved in the works Civil Defence were brought in to help search for and clear bodies to get the plant back into production. The net effect of the November raids was to destroy machine shops in the four-storey 1915 building, the original 1863 gunsmiths' building and nearby buildings, 1,600 machine tools, kill 53 employees, injure 89, 30 of them seriously and halt rifle production for three months. The Government Ministry of Supply and BSA immediately began a process of production dispersal throughout Britain, through the shadow factory scheme. Factories were set up at Tipton, Dudley, Smethwick, Blackheath, Lye, Kidderminster, Stourport, Tyseley, and Bromsgrove to manufacture Browning machine guns, Stoke, Corsham, and Newcastle-under-Lyme produced the Hispano cannon, Leicester and Studley Road produced the Besa machine gun, Ruislip produced the Oerlikon 20mm cannon, Stafford produced rocket projectiles, Tamworth produced two-pounder gun carriages, Mansfield produced the Boys Anti-tank gun and Shirley produced rifles. These were dispersal factories which were in addition to Small Heath and the other BSA factories opened in the two years following the 1940 blitz. At its peak Small Heath was running 67 factories engaged in war production. BSA operations were also dispersed to other companies under licence. In 1941 BSA was approached to produce a new pedal cycle with a maximum weight allowance of only 22 lb especially for airborne use. This required a new concept in frame design which BSA found, producing a machine which weighed 21 lb, one pound less than the design specification and which also exceeded the design requirement for an effective life of 50 miles many times over. Over 60,000 folding bicycles were produced, a figure equal to half the total production of military bicycles during World War II. BSA also produced folding motorcycles for the Airborne Division. In late 1942 BSA examined the Special Operations Executive designed Welgun with a view to manufacture. BSA were willing to manufacture the gun in the quantities required starting April 1943 but the cheaper and less accurate Sten Mk IV was adopted for production by the Ministry of Supply. BSA bought the Sunbeam motorcycles and bicycle business from Associated Motor Cycles Ltd in 1943 and then Ariel Motors Ltd in 1944. During the course of the conflict BSA produced 1,250,000 Lee–Enfield .303 service rifles, 404,383 Sten sub-machine guns, 468,098 Browning machine guns plus spares equivalent to another 100,000, 42,532 Hispano cannon, 32,971 Oerlikon cannon, 59,322 7.9 mm Besa machine guns, 3,218 15 mm Besa machine guns, 68,882 Boys Anti-tank guns, 126,334 motorcycles, 128,000 military bicycles (over 60,000 of which were folding paratrooper bicycles), 10,000,000 shell fuse cases, 3,485,335 magazines and 750,000 anti-aircraft rockets were supplied to the armed forces.

At the same time other parts of the Group were having similar problems. Before World War II Daimler had been linked with other Coventry motor manufacturers in a government-backed scheme for aero engine manufacture and had been allocated two shadow factories. Apart from this, BSA-owned Daimler was producing Scout Cars and Daimler Mk I Armoured Cars which had been designed by BSA at Small Heath not Coventry as well as gun turrets, gun parts, tank transmissions, rocket projectiles and other munitions. This activity had not gone unnoticed by the enemy, which made Radford Works a target in the Coventry air raids. Radford Works received direct hits in four separate air raids during 1940. None of these attacks were to seriously disrupt production, however two more serious air raids were carried out in April 1941 which destroyed half the factory. In all it is estimated that 170 bombs containing 52,000 lbs of explosive were dropped on Radford Works as well as the thousands of incendiaries. Like BSA, Daimler had to find dispersal units. A back-handed compliment was paid by Field Marshal Rommel to the workers at Radford Works when he used a captured Daimler Scout to escape following his defeat at El Alamein.

BSA cycles
BSA produced the first Sunbeam bicycle catalogue in 1949 and produced its own '4 Star' derailleur gear with an associated splined cassette hub and 4 sprocket cassette. This design was different from the 1930s Bayliss Wiley cassette hub which had a threaded sprocket carrier. BSA bought New Hudson motorcycle and bicycle business in 1950 and followed this up in 1951 with the purchase of Triumph Motorcycles which brought Jack Sangster onto the BSA board. The effect of this acquisition was to make BSA into the largest producer of motorcycles in the world at that time.

1952 saw BSA establish a Professional Cycling Team. Bob Maitland a successful amateur cyclist and the highest placed British finisher in the 1948 Olympic Games road race and now an independent rider in the BSA team was a BSA employee working in the design office as a draughtsman. It was Bob Maitland who was responsible for the design of post war BSA range of lightweight sports bicycles based on his knowledge of cycling. Bob Maitland also made some of the components used on the bicycles of the professional team which were not standard production machines. In the 1952 Tour of Britain Road Race run between Friday 22 August and Saturday 6 September, involving 14 individual stages and covering a total race distance of 1,470 miles, the BSA team of Bob Maitland, "Tiny" Thomas, Pete Proctor, Alf Newman and Stan Jones won the overall team race and Pete Proctor "King of the Mountains" classification. The riders also enjoyed success on the individual stages of the race. The team competed in four further events, 14 September Tour of the Chilterns, 1st "Tiny" Thomas and Team Prize, 21 September Weston-Super-Mare Grand Prix, Team Prize, 28 September Staffordshire Grand Prix, 1st Bob Maitland and Team Prize, 5 October Tour Revenge Race, Dublin, 1st "Tiny" Thomas and Team prize.

In 1953 BSA withdrew motorcycle production from BSA Cycles Ltd, the company it has established in 1919, by creating BSA Motorcycles Ltd.

In 1953 the BSA Professional Cycling Team was managed by Syd Cozens. Successes were 5/6 April Bournemouth Two Day Road Race, 1st Bob Maitland, 12 April Dover to London 63 Miles Road Race, 1st Stan Jones, 31 May Langsett 90 Miles Road Race, 1st Bob Maitland and "King of the Mountains", 7 June Tour of the Wrekin, 1st Bob Maitland, 12 July Severn Valley 100 Miles Road Race, 1st "Tiny" Thomas, 19 July Jackson Trophy, Newcastle, Team Prize, 9 August Les Adams Memorial 80 Miles Road Race, 1st Alf Newman, Team Prize, "King of the Mountains" Arthur Ilsley, 30 August Weston-Super-Mare 100 Miles Grand Prix, 1st Bob Maitland, Team Prize. The team also competed in the 1,624-mile, 12 stage, 1953 Tour of Britain Road Race. The 1953 line up had changed as Arthur Ilsley replaced Pete Proctor in the team. "Tiny" Thomas won the overall individual classification, the Team were runners-up in the team competition and Arthur Ilsley was 3rd in the "King of the Mountains" competition. Bob Maitland also had notable success by winning the Independent National Championship.

1954 saw the introduction of the BSA Quick Release 3 Speed hub gear. It was a split axle three speed gear intended for use with bicycles equipped with oil bath chainguards. The original BSA 3 speed hub gear had been made under licence from the Three-Speed Gear Syndicate since 1907. The design was later to be classified as the Sturmey-Archer 'Type X', but all BSA hub gear production ceased in 1955

Management
Sir Bernard Docker remained chairman of BSA until 1956 when the BSA removed him. In an acrimonious dispute conducted in the media the matter was brought to the BSA shareholders at the Annual General Meeting where the decision of the Board was upheld. Another significant departure for the fortune of the BSA Group but less controversial was the retirement on ill health grounds of James Leek CBE, managing director from 1939 until his retirement. Sir Bernard Docker was replaced as Chairman of the BSA Board by Jack Sangster.

Dispersals
The BSA bicycle division, BSA Cycles Ltd., including the BSA cycle dealer network was sold to Raleigh in 1957. Raleigh initially continued bicycle production in Birmingham at Coventry Road, Sheldon, Birmingham 26, into the early 1960s using up BSA parts but as time went on more stock Raleigh parts and fittings were used, some continuing to bear the 'piled arms' stamp. TI Group, owners of the British Cycle Corporation, bought Raleigh in 1960 thus gaining access to the BSA brand. Bicycles bearing the BSA name are currently manufactured and distributed within India by TI Cycles of India but have no direct connection to the original Birmingham BSA company.

In 1960, Daimler was sold to Jaguar.

1961 was the centenary year of the BSA Group and, in recognition of this milestone, the company magazine produced an anniversary issue of BSA Group News in June called BSA Centenary 1861–1961, in which many of the achievements of the Group were celebrated. This year also saw the end of military rifle production; however, BSA still continued to make sporting guns.

Bicycles
According to Charles Spencer, BSA was manufacturing the "Delta" bicycle c. 1869. In 1880 the company was approached to manufacture the "Otto Dicycle". An initial contract was signed to produce 210 and a further contract followed for a further 200. In all it is believed that a total of 953 Otto machines were made. BSA then went into bicycle production on their own account, the first machines to their own specification being exhibited at the 1881 Stanley Show. BSA went on to design and manufacture a "safety" bicycle (patent:15,342 of 1884). BSA was also producing tricycles and a licence was obtained in 1885 to manufacture ball bearings. BSA ceased bicycle manufacture in 1887 because of the demand for arms. Bicycle component manufacture commenced in 1894 and BSA continued to supply the bicycle trade up to 1936. The company recommenced bicycle manufacture on their own account again in 1908 and these were exhibited at the Stanley Show in 1909. Bicycle manufacture was what led BSA into motorcycles. BSA produced bicycles for both the police and military and notably a folding bicycle for the British Army during World War I and the more well known folding Paratroopers bicycle during World War II. BSA supplied the Irish Army with bicycles after 1922.

BSA manufactured a range of bicycles from utility roadsters through to racing bicycles. The BSA range of Sports bicycles expanded in the 1930s following the granting of a patent for a new lighter design of seat lug in 1929 and tandems were introduced into the BSA bicycle range as well. BSA had a reputation for quality and durability and their components were more expensive that either Chater-Lea or Brampton. BSA launched a high end club cyclists machine in the early 1930s initially branded as the "Super-eeze". Never slow to avail of publicity BSA sponsored the great Australian cyclist Hubert Opperman and re-branded the top of the range machine the "Opperman" model. A less expensive range of clubman lightweight machines was introduced from 1936 with the "Cyclo" 3 speed derailleur equipped "Clubman". Subtle changes were made to the range, most models being equipped with "Russ" patent forks and some models were made for only two seasons. This all stopped around September 1939 with the outbreak of war. A revised catalogue with a much reduced range was issued in March 1940 which also saw the launch of the BSA "Streamlight" model. A novel all white bicycle was produced for the blackout but had disappeared from a severely reduced bicycle range the details of which were circulated to dealers from December 1941. BSA had ceased production of their 3 speed hub gear in 1939 and production appears to have started again by 1945 although with a black finish instead of chromium plating. BSA bought Sunbeam in 1943 and produced Sunbeam bicycles using up existing frames and parts and using BSA components for the missing bits. The first BSA produced Sunbeam catalogue was published in 1949

Post war BSA expanded their bicycle range but faced problems of shortages of raw materials such as steel and was required to export a lot of their manufactured output in order to get a Government licence to purchase the necessary raw materials. The company moved bicycle production to the new Waverley Works after World War II. BSA continued to innovate introducing the 4 Star derailleur gear in 1949 along with an associated 4-speed 'unit' or cassette hub. The derailleur design was altered from 1950 and was certainly available up to 1953 but was not a great success. BSA bought New Hudson in 1950 and started to manufacture and sell New Hudson branded machines as well as Sunbeam. It appears that the top of the range BSA lightweight club cyclist machine was the "Gold Column" and this appears to have been changed into the BSA "Tour of Britain" model following the success of the BSA Professional Cycling Team in the 1952 Tour of Britain race. The "Tour of Britain" model was heavily promoted in the BSA 1953 sales literature. The factory made "Tour of Britain" model was not the same as those ridden by the professional team. Only eight machines were crafted for the professional team and none of the components appear to have been standard BSA parts. 1953 saw BSA separate the bicycle / motorcar and motorcycle business into different holdings.

The good times were coming to an end and demand for bicycles fell with the end of rationing in 1954. James Leek, managing director of BSA Cycles Ltd was suffering ill health and he retired in 1956, the same year the BSA Chairman, Sir Bernard Docker, was removed from the BSA Board. Jack Sangster who had joined the BSA Board in 1951 following the purchase of his company Triumph Motorcycles became chairman. The bicycle manufacturing business BSA Cycles Ltd was sold to Raleigh Industries in 1957.

Motorcycles
BSA Motorcycles were made by BSA Cycles Ltd, under the BSA parent, until 1953 when the motorcycle business was moved into holding BSA Motorcycles Ltd. The first instance of intention to produce motorcycles was reported in The Motor Cycle, a British motorcycling journal, in July 1906. The first wholly BSA motorcycle, the $3 1/2$ H.P. was built in 1910 and displayed at the first Olympia Show, London on 21 November in that year. Sir Hallewell Rogers, BSA Chairman, had informed the shareholders at the company's 1910 AGM in Birmingham "We have decided to put a motor-bicycle on the market for the coming season .... These machines will be on exhibit at the Cycle and Motor Show on November 21st, after which date we look forward to commencing delivery". The machines were available for the 1911 season and entire production sold out. BSA had previously acquired a commercially available engine in 1905 and fitted it to one of their bicycle frames and discovered at first hand the problems that needed to be overcome. BSA Cycles Ltd was set up as a subsidiary company in 1919 under Managing Director Charles Hyde to manufacture both bicycles and motorcycles.

Civilian firearms

 * The 1906 war office pattern rifle
 * The Sportsman series of .22 Long Rifle bolt-action rifles
 * Various Martini action target .22lr rifles
 * The Ralock and Armatic semi automatic .22lr rifles
 * Various bolt action hunting rifles

Trademarks
Rights to the motorcycle brand went to Norton Villiers Triumph and on its liquidation were purchased by BSA Company. Rights to the guns brand were acquired by Gamo for its new subsidiary, BSA Guns (UK) Limited.

In popular culture
The Birmingham Small Arms Company (BSA) plays a prominent role in the inaugural season of the British crime drama series Peaky Blinders, as well as in subsequent seasons, where it serves as a focal point for armaments production, and as the day-to-day workplace of a number of characters.

title: Birmingham Small Arms Company


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Bovril

Bovril is the trademarked name of a thick and salty meat extract paste, similar to a yeast extract, developed in the 1870s by John Lawson Johnston. It is sold in a distinctive bulbous jar and as cubes and granules. Bovril is owned and distributed by Unilever UK. Its appearance is similar to the British Marmite and its Australian equivalent Vegemite; however, unlike these products, Bovril is not vegetarian.

Bovril can be made into a drink (referred to in the UK as a "beef tea") by diluting with hot water or, less commonly, with milk. It can be used as a flavouring for soups, broth, stews or porridge, or as a spread, especially on toast in a similar fashion to Marmite and Vegemite.

Etymology

The first part of the product's name comes from Latin bovīnus, meaning "ox". Johnston took the -vril suffix from Edward Bulwer-Lytton's then-popular novel, The Coming Race (1871), the plot of which revolves around a superior race of people, the Vril-ya, who derive their powers from an electromagnetic substance named "Vril". Therefore, Bovril indicates great strength obtained from an ox.

History
In 1870, in the Franco-Prussian War, Napoleon III ordered one million cans of beef to feed his troops. The task of providing this went to John Lawson Johnston, a Scottish butcher living in Canada. Large quantities of beef were available across the British Dominions and South America, but transport and storage were problematic. Therefore, Johnston created a product known as 'Johnston's Fluid Beef', later called Bovril, to meet Napoleon's needs. By 1888, over 3,000 UK public houses, grocers and dispensing chemists were selling Bovril. In 1889, Bovril Ltd was formed to develop Johnston's business further.

During the 1900 Siege of Ladysmith in the Second Boer War, a Bovril-like paste was produced from horsemeat within the garrison. Nicknamed Chevril (a portmanteau of cheval, French for horse, and Bovril) it was made by boiling down horse or mule meat to a jelly and serving it as a beef tea-like mixture. Bovril also produced concentrated, pemmican-like dried beef as part of the British Army emergency field ration during the war. The ration came in the form of a pocket-sized tin can that contained the beef on one half alongside a dried cocoa drink on another half. The dried beef could be eaten alone, or mixed with water to create a beef tea.

Bovril continued to function as a "war food" in World War I and was frequently mentioned in the 1930 account Not So Quiet: Stepdaughters of War by Helen Zenna Smith. It describes the drink being prepared for the casualties at Mons where "the orderlies were just beginning to make Bovril for the wounded, when the bearers and ambulance wagons were shelled as they were bringing the wounded into the hospital".

When John Lawson Johnston died, his son George Lawson Johnston inherited and took over the Bovril business. In 1929, George Lawson Johnston was made Baron Luke, of Pavenham, in the county of Bedford.

Bovril's instant beef stock was launched in 1966 and its "King of Beef" range of instant flavours for stews, casseroles and gravy in 1971. In 1971, James Goldsmith's Cavenham Foods acquired the Bovril Company but then sold most of its dairies and South American operations to finance further take-overs. The brand is now owned by the Anglo-Dutch multinational Unilever, which bought Bovril in 2001.

In 2004, Unilever removed beef ingredients from the Bovril formula, rendering it vegetarian. This was mainly due to concerns about decreasing sales, particularly from exports due to an export ban on British beef, as a result of the growing popularity of vegetarianism, religious dietary requirements, and public concerns about bovine spongiform encephalopathy. In 2006, Unilever reversed that decision and reintroduced beef ingredients to their Bovril formula once sales increased and the beef export bans were lifted. Unilever now produces Bovril using beef extract and a chicken variety using chicken extract.

In November 2020, Forest Green Rovers Football Club announced a collaboration with the makers of Bovril to create a beet-based version of Bovril to be sold at their New Lawn stadium, where meat-based products had been removed from sale some years prior.

Licensed production
Bovril is produced in South Africa by the Bokomo division of Pioneer Foods.

Cultural significance

Bovril was promoted as a superfood in the early 20th century. Advertisements recommended people to dilute it into a tea or spread it on their morning toast. Some adverts even claimed that Bovril could protect one from influenza.

Bovril jars are commonly excavated as part of archaeological assemblages, such as at Knowles Mill in Worcestershire.

Since its invention, Bovril has become an icon of British culture. It is associated with football culture. During the winter, British football fans in stadium terraces drink it as a tea from Thermos flasks – or from disposable cups in Scotland, where thermoses are banned from football stadiums. Bovril holds the unusual distinction of having been advertised with a Pope. An advertising campaign of the early 20th century in Britain depicted Pope Leo XIII seated on his throne, bearing a mug of Bovril. The campaign slogan read: The Two Infallible Powers – The Pope & Bovril.

Bovril beef tea was the only hot drink that Ernest Shackleton's team had when they were marooned on Elephant Island during the 1914–1917 Endurance Expedition.

In the film In Which We Serve, the officers on the bridge are served "Bovril rather heavily laced with sherry" to warm them up, after being rescued during the Dunkirk evacuation of the British Expeditionary Force.

British mountaineer Chris Bonington appeared in TV commercials for Bovril in the 1970s and 1980s in which he recalled melting snow and ice on Everest to make hot drinks.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Bovril


text:
Benjamin D. Santer

Benjamin David Santer (born June 3, 1955) was a climate researcher at Lawrence Livermore National Laboratory and former researcher at the University of East Anglia's Climatic Research Unit. He retired from the Lawrence Livermore National Laboratory in 2021 and is now a Distinguished Scholar in Residence at Woods Hole Oceanographic Institution. He also worked at the Max Planck Institute for Meteorology from 1987 to 1992. He specializes mainly in statistical analysis of climate data sets, and detection/attribution of climate change forcings.

Since 2012, Santer has been listed on the board of directors of the National Center for Science Education.

Honors
Santer received a B.Sc. in Environmental Sciences and a 1987 Ph.D. in Climatology from the Climatic Research Unit of the University of East Anglia.

In 1998 Santer was awarded a MacArthur Fellowship for research supporting the finding that human activity contributes to global warming. He has also received the Ernest Orlando Lawrence Award and a Distinguished Scientist Fellowship from the U.S. Department of Energy and the Norbert Gerbier/MUMM award from the World Meteorological Organization. He ranked twelfth amongst climate scientists in a 2002 assessment of most cited scientists in the field of global warming.

In 2011, Santer was elected as a fellow of the American Geophysical Union and as a member of the National Academy of Sciences.

1995 AR2 Chapter 8
Santer was the convening Lead Author of Chapter 8 of 1995 IPCC Working Group I Report (AR2 WGI), which addressed the global warming issue.

In a June 12, 1996 editorial-page piece in The Wall Street Journal, Frederick Seitz, chair of the George C. Marshall Institute and Science and Environmental Policy Project, claimed that alterations made to Chapter 8 of the 1995 IPCC report were made to "deceive policy makers and the public into believing that the scientific evidence shows human activities are causing global warming." Similar charges were made by the Global Climate Coalition (GCC), a consortium of industry interests; specifically, they accused Santer of "scientific cleansing."

Santer and 40 other scientists responded to The Wall Street Journal that all IPCC procedural rules were followed, and that IPCC procedures required changes to the draft in response to comments from governments, individual scientists, and non-governmental organizations. They stated that the pre- and post-Madrid versions of Chapter 8 were equally cautious in their statements; that roughly 20% of Chapter 8 is devoted to the discussion of uncertainties in estimates of natural climate variability and the expected signal due to human activities; and that both versions of the chapter reached the same conclusion: "Taken together, these results point towards a human influence on climate."

Gold Standard Paper
On February 25, 2019 Santer et al. published the paper Celebrating the anniversary of three key events in climate change science in Nature Climate Change, claiming to have reached the 5-sigma "gold standard level" of statistical proof of human influence in global climate change using three sets of satellite data.

title: Benjamin D. Santer


text:
Bernoulli number

In mathematics, the Bernoulli numbers $B± n$ are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by $$B^{-{}}_n$$ and $$B^{+{}}_n$$; they differ only for $B_{n}$, where $$B^{-{}}_1=-1/2$$ and $$B^{+{}}_1=+1/2$$. For every odd $n = 1$, $n > 1$. For every even $B_{n} = 0$, $n > 0$ is negative if $B_{n}$ is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials $$B_n(x)$$, with $$B^{-{}}_n=B_n(0)$$ and $$B^+_n=B_n(1)$$.

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm prepared by Babbage for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

Notation
The superscript $n$ used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the $±$ term is affected:
 * $n = 1$ with $B− n$ ( / ) is the sign convention prescribed by NIST and most modern textbooks.
 * $B− 1 = −1⁄2$ with $B+ n$ ( / ) was used in the older literature, and (since 2022) by Donald Knuth following Peter Luschny's "Bernoulli Manifesto".

In the formulas below, one can switch from one sign convention to the other with the relation $$B_n^{+}=(-1)^n B_n^{-}$$, or for integer $n$ = 2 or greater, simply ignore it.

Since $B+ 1 = +1⁄2$ for all odd $Bn = 0$, and many formulas only involve even-index Bernoulli numbers, a few authors write "$n > 1$" instead of $Bn$. This article does not follow that notation.

Early history
The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.

Methods to calculate the sum of the first $1⁄2$ positive integers, the sum of the squares and of the cubes of the first $1⁄6$ positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras (c. 572–497 BCE, Greece), Archimedes (287–212 BCE, Italy), Aryabhata (b. 476, India), Abu Bakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham (965–1039, Iraq).

During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West Thomas Harriot (1560–1621) of England, Johann Faulhaber (1580–1635) of Germany, Pierre de Fermat (1601–1665) and fellow French mathematician Blaise Pascal (1623–1662) all played important roles.

Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae, far higher than anyone before him, but he did not give a general formula.

Blaise Pascal in 1654 proved Pascal's identity relating the sums of the $B2n$th powers of the first $p$ positive integers for $n$.

The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants $p = 0, 1, 2, ..., k$ which provide a uniform formula for all sums of powers.

The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the $1⁄30$th powers for any positive integer $B_{0}, B_{1}, B_{2},...$ can be seen from his comment. He wrote:

 * "With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500."

Bernoulli's result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712. However, Seki did not present his method as a formula based on a sequence of constants.

Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of Abraham de Moivre.

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834. Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on):

 * "Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants $c$ ... would provide a uniform
 * $\sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1} 1 B_1 n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right) $
 * for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for $B_{0}, B_{1}, B_{2},$ from polynomials in $1⁄42$ to polynomials in $1⁄30$."

In the above Knuth meant $$B_1^-$$; instead using $$B_1^+$$ the formula avoids subtraction:
 * $ \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}+\binom{m+1} 1 B^+_1 n^m+\binom{m+1} 2B_2n^{m-1}+\cdots+\binom{m+1}mB_mn\right). $

Reconstruction of "Summae Potestatum"

The Bernoulli numbers (n)/(n) were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted $Σ n^{m}$, $A$, $B$ and $C$ by Bernoulli are mapped to the notation which is now prevalent as $D$, $A = B_{2}$, $B = B_{4}$, $C = B_{6}$. The expression $D = B_{8}$ means $c·c−1·c−2·c−3$ – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers $c·(c−1)·(c−2)·(c−3)$. The factorial notation $c^$ as a shortcut for $k!$ was not introduced until 100 years later. The integral symbol on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter $1 × 2 × ... × k$ for "summa" (sum). The letter $S$ on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as $n$. Putting things together, for positive $1, 2, ..., n$, today a mathematician is likely to write Bernoulli's formula as:

 * $$ \sum_{k=1}^n k^c = \frac{n^{c+1}}{c+1}+\frac 1 2 n^c+\sum_{k=2}^c \frac{B_k}{k!} c^{\underline{k-1}}n^{c-k+1}.$$

This formula suggests setting $c$ when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the falling factorial $B_{1} = 1⁄2$ has for $c^$ the value $k = 0$. Thus Bernoulli's formula can be written
 * $$ \sum_{k=1}^n k^c = \sum_{k=0}^c \frac{B_k}{k!}c^{\underline{k-1}} n^{c-k+1}$$

if $1⁄c + 1$, recapturing the value Bernoulli gave to the coefficient at that position.

The formula for $$\textstyle \sum_{k=1}^n k^9$$ in the first half of the quotation by Bernoulli above contains an error at the last term; it should be $$-\tfrac {3}{20}n^2$$ instead of $$-\tfrac {1}{12}n^2$$.

Definitions
Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned:

 * a recursive equation,
 * an explicit formula,
 * a generating function,
 * an integral expression.

For the proof of the equivalence of the four approaches.

Recursive definition
The Bernoulli numbers obey the sum formulas
 * $$ \begin{align} \sum_{k=0}^{m}\binom {m+1} k B^{-{}}_k &= \delta_{m, 0} \\ \sum_{k=0}^{m}\binom {m+1} k B^{+{}}_k &= m+1 \end{align}$$

where $$m=0,1,2...$$ and $B_{1} = 1/2$ denotes the Kronecker delta. Solving for $$B^{\mp{}}_m$$ gives the recursive formulas
 * $$\begin{align}

B_m^{-{}} &= \delta_{m, 0} - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^{-{}}_k}{m - k + 1} \\ B_m^+ &= 1 - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^+_k}{m - k + 1}. \end{align}$$

Explicit definition
In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers, usually giving some reference in the older literature. One of them is (for $$m\geq 1$$):
 * $$\begin{align}

B^-_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j j^m \\ B^+_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j (j + 1)^m. \end{align}$$

Generating function
The exponential generating functions are
 * $$\begin{alignat}{3}

\frac{t}{e^t - 1}   &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} -1 \right) &&= \sum_{m=0}^\infty \frac{B^{-{}}_m t^m}{m!}\\ \frac{t}{1 - e^{-t}} &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} +1 \right) &&= \sum_{m=0}^\infty \frac{B^+_m t^m}{m!}. \end{alignat}$$ where the substitution is $$t \to - t$$.

If we let $$F(t)=\sum_{i=1}^\infty f_it^i$$ and $$G(t)=1/(1+F(t))=\sum_{i=0}^\infty g_it^i$$ then

 * $$G(t)=1-F(t)G(t).$$

Then $$g_0=1$$ and for $$m>0$$ the m$5⁄66$ term in the series for $$G(t)$$ is:

 * $$g_mt^i=-\sum_{j=0}^{m-1}f_{m-j}g_jt^m$$

If

 * $$F(t)=\frac{e^t-1}t-1=\sum_{i=1}^\infty \frac{t^i}{(i+1)!}$$

then we find that

 * $$G(t)=t/(e^t-1)$$

 * $$\begin{align}

m!g_m&=-\sum_{j=0}^{m-1}\frac{m!}{j!}\frac{j!g_j}{(m-j+1)!}\\ &=-\frac 1{m+1}\sum_{j=0}^{m-1}\binom{m+1}jj!g_j\\ \end{align}$$

showing that the values of $$i!g_i$$ obey the recursive formula for the Bernoulli numbers $$B^-_i$$.

The (ordinary) generating function
 * $$ z^{-1} \psi_1(z^{-1}) = \sum_{m=0}^{\infty} B^+_m z^m$$

is an asymptotic series. It contains the trigamma function $δ$.

Integral Expression
From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers:

 * $$B_{2n} = 4n (-1)^{n+1} \int_0^{\infty} \frac{t^{2n-1}}{e^{2 \pi t} -1 } \mathrm{d} t $$

Bernoulli numbers and the Riemann zeta function

The Bernoulli numbers can be expressed in terms of the Riemann zeta function:

 * $ψ_{1}$          for $B+ n = −nζ(1 − n)$.

Here the argument of the zeta function is 0 or negative. As $$\zeta(k)$$ is zero for negative even integers (the trivial zeroes), if n>1 is odd, $$\zeta(1-n)$$ is zero.

By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:

 * $$ B_{2n} = \frac {(-1)^{n+1}2(2n)!} {(2\pi)^{2n}} \zeta(2n) \quad $$ for $n ≥ 1$.

Now the argument of the zeta function is positive.

It then follows from $n ≥ 1$ ($ζ → 1$) and Stirling's formula that
 * $$ |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n} \quad $$ for $n → ∞$.

Efficient computation of Bernoulli numbers
In some applications it is useful to be able to compute the Bernoulli numbers $n → ∞$ through $B_{0}$ modulo $691⁄2730$, where $7⁄6$ is a prime; for example to test whether Vandiver's conjecture holds for $3617⁄510$, or even just to determine whether $43867⁄798$ is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) $B_{p − 3}$ arithmetic operations would be required. Fortunately, faster methods have been developed which require only $p^{2}$ operations (see big $174611⁄330$ notation).

David Harvey describes an algorithm for computing Bernoulli numbers by computing $O(p (log p)^{2})$ modulo $n$ for many small primes $n$, and then reconstructing $B_{n}$ via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is $B_{n}$ and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed $O(n^{2} log(n)^{2 + ε})$ for $B_{n}$. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner computed $n = 10^{8}$ to full precision for $B_{n}$ in December 2002 and Oleksandr Pavlyk for $n = 10^{6}$ with Mathematica in April 2008.

 * {| class="wikitable defaultright col1left"

! Computer !! Year !! n !! Digits*
 * J. Bernoulli || ~1689 || 10 || 1
 * L. Euler || 1748 || 30 || 8
 * J. C. Adams || 1878 || 62 || 36
 * D. E. Knuth, T. J. Buckholtz || 1967 || $n$ || $c$
 * G. Fee, S. Plouffe || 1996 || $N$ || $n$
 * G. Fee, S. Plouffe || 1996 || $th$ || $p$
 * B. C. Kellner || 2002 || $p$ || $p$
 * O. Pavlyk || 2008 || $p$ || $O$
 * D. Harvey || 2008 || $p$ || $p$
 * }
 * * Digits is to be understood as the exponent of 10 when $n = 10^{7}$ is written as a real number in normalized scientific notation.
 * G. Fee, S. Plouffe || 1996 || $1,672$ || $3,330$
 * B. C. Kellner || 2002 || $10,000$ || $27,677$
 * O. Pavlyk || 2008 || $100,000$ || $376,755$
 * D. Harvey || 2008 || $1,000,000$ || $4,767,529$
 * }
 * * Digits is to be understood as the exponent of 10 when $B_{n}$ is written as a real number in normalized scientific notation.
 * D. Harvey || 2008 || $10,000,000$ || $57,675,260$
 * }
 * * Digits is to be understood as the exponent of 10 when $B− 1 = −1⁄2$ is written as a real number in normalized scientific notation.

A possible algorithm for computing Bernoulli numbers in the Julia programming language is given by

Asymptotic analysis
Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula. Assuming that $100,000,000$ is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as

 * $$\sum_{k=a}^{b-1} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^-_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_-(f,m).$$

This formulation assumes the convention $B+ 1 = +1⁄2$. Using the convention $s^{\overline{k}}|undefined$ the formula becomes

 * $$\sum_{k=a+1}^{b} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^+_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_+(f,m).$$

Here $$f^{(0)}=f$$ (i.e. the zeroth-order derivative of $$f$$ is just $$f$$). Moreover, let $$f^{(-1)}$$ denote an antiderivative of $$f$$. By the fundamental theorem of calculus,

 * $$\int_a^b f(x)\,dx = f^{(-1)}(b) - f^{(-1)}(a).$$

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

 * $$ \sum_{k=a+1}^{b} f(k)= \sum_{k=0}^m \frac{B_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m). $$

This form is for example the source for the important Euler–Maclaurin expansion of the zeta function

 * $$ \begin{align}

\zeta(s) & =\sum_{k=0}^m \frac{B^+_k}{k!} s^{\overline{k-1}} + R(s,m) \\ & = \frac{B_0}{0!}s^{\overline{-1}} + \frac{B^+_1}{1!} s^{\overline{0}} + \frac{B_2}{2!} s^{\overline{1}} +\cdots+R(s,m) \\ & = \frac{1}{s-1} + \frac{1}{2} + \frac{1}{12}s + \cdots + R(s,m). \end{align} $$

Here $ψ$ denotes the rising factorial power.

Bernoulli numbers are also frequently used in other kinds of asymptotic expansions. The following example is the classical Poincaré-type asymptotic expansion of the digamma function $m$.

 * $$\psi(z) \sim \ln z - \sum_{k=1}^\infty \frac{B^+_k}{k z^k} $$

Sum of powers
Bernoulli numbers feature prominently in the closed form expression of the sum of the $n$th powers of the first $m, n ≥ 0$ positive integers. For $n$ define

 * $$S_m(n) = \sum_{k=1}^n k^m = 1^m + 2^m + \cdots + n^m. $$

This expression can always be rewritten as a polynomial in $m + 1$ of degree $( m + 1 k )$. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:
 * $$S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m \binom{m + 1}{k} B^+_k n^{m + 1 - k} = m! \sum_{k=0}^m \frac{B^+_k n^{m + 1 - k}}{k! (m+1-k)!} ,$$

where $m$ denotes the binomial coefficient.

For example, taking $0, 1, 3, 6, ...$ to be 1 gives the triangular numbers $m$.

 * $$ 1 + 2 + \cdots + n = \frac{1}{2} (B_0 n^2 + 2 B^+_1 n^1) = \tfrac12 (n^2 + n).$$

Taking $0, 1, 5, 14, ...$ to be 2 gives the square pyramidal numbers $(4n − 1)$.

 * $$1^2 + 2^2 + \cdots + n^2 = \frac{1}{3} (B_0 n^3 + 3 B^+_1 n^2 + 3 B_2 n^1) = \tfrac13 \left(n^3 + \tfrac32 n^2 + \tfrac12 n\right).$$

Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
 * $$S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m (-1)^k \binom{m + 1}{k} B^{-{}}_k n^{m + 1 - k}.$$

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sums of powers.

Faulhaber's formula was generalized by V. Guo and J. Zeng to a $676,752,569$-analog.

Taylor series
The Bernoulli numbers appear in the Taylor series expansion of many trigonometric functions and hyperbolic functions.

$$\begin{align} \tan x &= \hphantom \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} }{(2n)!}\; x^{2n-1}, && \left|x \right| < \frac \pi 2. \\ \cot x &= {1\over x} \sum_{n=0}^\infty \frac{(-1)^n B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi. \\ \tanh x &= \hphantom \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\;x^{2n-1}, && |x| < \frac \pi 2. \\ \coth x &= {1\over x} \sum_{n=0}^\infty \frac{B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi. \end{align}$$

Laurent series
The Bernoulli numbers appear in the following Laurent series:

Digamma function: $$ \psi(z)= \ln z- \sum_{k=1}^\infty \frac {B_k^{+{}}} {k z^k} $$

Use in topology
The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic $ES_{n}$-spheres which bound parallelizable manifolds involves Bernoulli numbers. Let $n ≥ 2$ be the number of such exotic spheres for $n!$, then

 * $$\textit{ES}_n = (2^{2n-2}-2^{4n-3}) \operatorname{Numerator}\left(\frac{B_{4n}}{4n} \right) .$$

The Hirzebruch signature theorem for the $f$ genus of a smooth oriented closed manifold of dimension 4n also involves Bernoulli numbers.

Connections with combinatorial numbers
The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the inclusion–exclusion principle.

Connection with Worpitzky numbers
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function $k^{m}$ and the power function $B_{0} = 1$ is employed. The signless Worpitzky numbers are defined as

 * $$ W_{n,k}=\sum_{v=0}^k (-1)^{v+k} (v+1)^n \frac{k!}{v!(k-v)!} . $$

They can also be expressed through the Stirling numbers of the second kind

 * $$ W_{n,k}=k! \left\{ {n+1\atop k+1} \right\}.$$

A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the harmonic sequence 1, $q$, $L$, ...

 * $$ B_{n}=\sum_{k=0}^n (-1)^k \frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^n \frac{1}{k+1} \sum_{v=0}^k (-1)^v (v+1)^n {k \choose v}\ . $$

This representation has $B_{1} = 1 − 1⁄2$.

Consider the sequence $B_{2} = 1 − 3⁄2 + 2⁄3$, $B_{3} = 1 − 7⁄2 + 12⁄3 − 6⁄4$. From Worpitzky's numbers, applied to $B_{4} = 1 − 15⁄2 + 50⁄3 − 60⁄4 + 24⁄5$ is identical to the Akiyama–Tanigawa transform applied to $B_{5} = 1 − 31⁄2 + 180⁄3 − 390⁄4 + 360⁄5 − 120⁄6$ (see Connection with Stirling numbers of the first kind). This can be seen via the table:

 * {| style="text-align:center"

 * + Identity of Worpitzky's representation and Akiyama–Tanigawa transform
 * 1|| || || || || ||0||1|| || || || ||0||0||1|| || || ||0||0||0||1|| || ||0||0||0||0||1||
 * 1||−1|| || || || ||0||2||−2|| || || ||0||0||3||−3|| || ||0||0||0||4||−4|| || || || || || ||
 * 1||−3||2|| || || ||0||4||−10||6|| || ||0||0||9||−21||12|| || || || || || || || || || || || ||
 * 1||−7||12||−6|| || ||0||8||−38||54||−24|| || || || || || || || || || || || || || || || || || ||
 * 1||−15||50||−60||24|| || || || || || || || || || || || || || || || || || || || || || || || ||
 * }
 * 1||−7||12||−6|| || ||0||8||−38||54||−24|| || || || || || || || || || || || || || || || || || ||
 * 1||−15||50||−60||24|| || || || || || || || || || || || || || || || || || || || || || || || ||
 * }
 * 1||−15||50||−60||24|| || || || || || || || || || || || || || || || || || || || || || || || ||
 * }
 * }

The first row represents $B_{6} = 1 − 63⁄2 + 602⁄3 − 2100⁄4 + 3360⁄5 − 2520⁄6 + 720⁄7$.

Hence for the second fractional Euler numbers ($B+ 1 = +1⁄2$) /  ($s_{n}$):

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for $n ≥ 0$

 * $$ B_n=\frac n {2^{n+1}-2}\sum_{k=0}^{n-1} (-2)^{-k}\, W_{n-1,k} . $$

The simplified second Worpitzky's representation of the second Bernoulli numbers is:

($s_{0}, s_{0}, s_{1}, s_{0}, s_{1}, s_{2}, s_{0}, s_{1}, s_{2}, s_{3}, ...$) / ($s_{n}$) = $s_{0}, s_{1}, s_{2}, s_{3}, s_{4}$ × ($n$) / ($n + 1$)

which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:

The numerators of the first parentheses are (see Connection with Stirling numbers of the first kind).

Connection with Stirling numbers of the second kind
If one defines the Bernoulli polynomials $E_{0} = 1$ as:

 * $$ B_k(j)=k\sum_{m=0}^{k-1}\binom{j}{m+1}S(k-1,m)m!+B_k $$

where $E_{1} = 1 − 1⁄2$ for $E_{2} = 1 − 3⁄2 + 2⁄4$ are the Bernoulli numbers.

One also has the following for Bernoulli polynomials,

 * $$ B_k(j)=\sum_{n=0}^k \binom{k}{n} B_n j^{k-n}. $$

The coefficient of $1⁄2$ in $E_{3} = 1 − 7⁄2 + 12⁄4 − 6⁄8$ is $E_{4} = 1 − 15⁄2 + 50⁄4 − 60⁄8 + 24⁄16$.

Comparing the coefficient of $1⁄3$ in the two expressions of Bernoulli polynomials, one has:

 * $$ B_k=\sum_{m=0}^{k-1} (-1)^m \frac{m!}{m+1} S(k-1,m)$$

(resulting in $E_{5} = 1 − 31⁄2 + 180⁄4 − 390⁄8 + 360⁄16 − 120⁄32$) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.

Connection with Stirling numbers of the first kind
The two main formulas relating the unsigned Stirling numbers of the first kind $E_{6} = 1 − 63⁄2 + 602⁄4 − 2100⁄8 + 3360⁄16 − 2520⁄32 + 720⁄64$ to the Bernoulli numbers (with $n ≥ 1$) are

 * $$ \frac{1}{m!}\sum_{k=0}^m (-1)^{k} \left[{m+1\atop k+1}\right] B_k = \frac{1}{m+1}, $$

and the inversion of this sum (for $n + 1$, $n + 1$)

 * $$ \frac{1}{m!}\sum_{k=0}^m (-1)^k \left[{m+1\atop k+1}\right] B_{n+k} = A_{n,m}. $$

Here the number $n + 1⁄2^{n + 2} − 2$ are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.

 * {| class="wikitable" style="text-align:center"

! !!0!!1!!2!!3!!4 ! 0 ! 1 ! 2 ! 3 ! 4
 * + Akiyama–Tanigawa number
 * 1 || $j$ || $j$ || $n$ || $m$
 * 0 || $1⁄2$ || ... || ... || ...
 * }
 * }

The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See /.

An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes =, the autosequence is of the first kind. Example:, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: /, the second Bernoulli numbers (see ). The Akiyama–Tanigawa transform applied to $n$ = 1/ leads to (n) /  (n + 1). Hence:

 * {| class="wikitable" style="text-align:center"

! !! 0 !! 1 !! 2 !! 3 !! 4 ! 0 ! 1 ! 2 ! 3 ! 4
 * + Akiyama–Tanigawa transform for the second Euler numbers
 * 1 || $1⁄3$ || $1⁄4$ || $1⁄5$ || $1⁄2$
 * 0 || $1⁄3$ || $1⁄4$ || ... || ...
 * 0 || ... || ... || ... || ...
 * }

See and. ($n + 1$) / ($1⁄2, 1⁄6, 0, −1⁄30, 0, 1⁄42, ... = (1⁄2, 1⁄3, 3⁄14, 2⁄15, 5⁄62, 1⁄21, ...) × (1, 1⁄2, 0, −1⁄4, 0, 1⁄2, ...)$) are the second (fractional) Euler numbers and an autosequence of the second kind.

Also valuable for /  (see Connection with Worpitzky numbers).

Connection with Pascal's triangle
There are formulas connecting Pascal's triangle to Bernoulli numbers
 * $$ B^{+}_n=\frac{|A_n|}{(n+1)!}$$

where $$|A_n|$$ is the determinant of a n-by-n Hessenberg matrix part of Pascal's triangle whose elements are: $$ a_{i, k} = \begin{cases} 0 & \text{if } k>1+i \\ {i+1 \choose k-1} & \text{otherwise} \end{cases} $$

Example:

 * $$ B^{+}_6 =\frac{\det\begin{pmatrix}

1& 2& 0& 0& 0& 0\\ 1& 3& 3& 0& 0& 0\\ 1& 4& 6& 4& 0& 0\\ 1& 5& 10& 10& 5& 0\\ 1& 6& 15& 20& 15& 6\\ 1& 7& 21& 35& 35& 21 \end{pmatrix}}{7!}=\frac{120}{5040}=\frac 1 {42} $$

Connection with Eulerian numbers
There are formulas connecting Eulerian numbers $B_{k}(j)$ to Bernoulli numbers:

 * $$\begin{align}

\sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle &= 2^{n+1} (2^{n+1}-1) \frac{B_{n+1}}{n+1}, \\ \sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle \binom{n}{m}^{-1} &= (n+1) B_n. \end{align}$$

Both formulae are valid for $B_{k}$ if $k = 0, 1, 2,...$ is set to $1⁄5$. If $( j m + 1 )$ is set to −$1⁄6$ they are valid only for $(−1)^{m}⁄m + 1$ and $B_{1} = +1⁄2$ respectively.

A binary tree representation
The Stirling polynomials $[ n m ]$ are related to the Bernoulli numbers by $B_{1} = +1⁄2$. S. C. Woon described an algorithm to compute $n ≥ 0$ as a binary tree:

 * [[File:SCWoonTree.png]]

Woon's recursive algorithm (for $m ≥ 0$) starts by assigning to the root node $A_{n,m}$. Given a node $2^{−n}$ of the tree, the left child of the node is $n$ and the right child $n + 1$. A node $n + 2$ is written as $n + 2$ in the initial part of the tree represented above with ± denoting the sign of $1⁄6, 0, −1⁄30, 0, 1⁄42, ...$.

Given a node $1⁄6$ the factorial of $3⁄20$ is defined as

 * $$ N! = a_1 \prod_{k=2}^{\operatorname{length}(N)} a_k!. $$

Restricted to the nodes $1⁄30$ of a fixed tree-level $1⁄30$ the sum of $2^{n + 3} − 2⁄n + 2$ is $3, 14⁄3, 15⁄2, 62⁄5, 21, ...$, thus

 * $$ B_n = \sum_\stackrel{N \text{ node of}}{\text{ tree-level } n} \frac{n!}{N!}. $$

For example:

Integral representation and continuation
The integral
 * $$ b(s) = 2e^{s i \pi/2}\int_0^\infty \frac{st^s}{1-e^{2\pi t}} \frac{dt}{t} = \frac{s!}{2^{s-1}}\frac{\zeta(s)}{{  }\pi^s{  }}(-i)^s= \frac{2s!\zeta(s)}{(2\pi i)^s}$$

has as special values $n + 1$ for $n + 2$.

For example, $1⁄2, 0, −1⁄4, 0, 1⁄2, ...$ and $⟨ n m ⟩$. Here, $n$ is the Riemann zeta function, and $m$ is the imaginary unit. Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated

 * $$ \begin{align}

p &= \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots \right) = 0.0581522\ldots \\ q &= \frac{15}{2\pi^5}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+\cdots \right) = 0.0254132\ldots \end{align}$$

Another similar integral representation is
 * $$ b(s) = -\frac{e^{s i \pi/2}}{2^{s}-1}\int_0^\infty \frac{st^{s}}{\sinh\pi t} \frac{dt}{t}= \frac{2e^{s i \pi/2}}{2^{s}-1}\int_0^\infty \frac{e^{\pi t}st^s}{1-e^{2\pi t}} \frac{dt}{t}. $$

The relation to the Euler numbers and $\pi$
The Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers $n ≥ 0$ are in magnitude approximately $B_{1}$ times larger than the Bernoulli numbers $B_{1}$. In consequence:

 * $$ \pi \sim 2 (2^{2n} - 4^{2n}) \frac{B_{2n}}{E_{2n}}. $$

This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations.

Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd $1⁄2$, $n ≥ 1$ (with the exception $n ≥ 2$), it suffices to consider the case when $1⁄4$ is even.

 * $$\begin{align}

B_n &= \sum_{k=0}^{n-1}\binom{n-1}{k} \frac{n}{4^n-2^n}E_k & n&=2, 4, 6, \ldots \\[6pt] E_n &= \sum_{k=1}^n \binom{n}{k-1} \frac{2^k-4^k}{k} B_k & n&=2,4,6,\ldots \end{align}$$

These conversion formulas express a connection between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to π. These numbers are defined for $σ_{n}(x)$ as

 * $$ S_n = 2 \left(\frac{2}{\pi}\right)^n \sum_{k = 0}^\infty \frac{ (-1)^{kn} }{(2k+1)^n} = 2 \left(\frac{2}{\pi}\right)^n \lim_{K\to \infty} \sum_{k = -K}^K (4k+1)^{-n}. $$

The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by Leonhard Euler in a landmark paper De summis serierum reciprocarum (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few of these numbers are

 * $$ S_n = 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots $$ ( / )

These are the coefficients in the expansion of $B_{n} = n!σ_{n}(1)$.

The Bernoulli numbers and Euler numbers can be understood as special views of these numbers, selected from the sequence $σ_{n}(1)$ and scaled for use in special applications.

 * $$\begin{align}

B_{n} &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] \frac{n! }{2^n - 4^n}\, S_{n}\, & n&= 2, 3, \ldots \\ E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] n! \, S_{n+1} & n &= 0, 1, \ldots \end{align}$$

The expression [$n ≥ 1$ even] has the value 1 if $N = [1,2]$ is even and 0 otherwise (Iverson bracket).

These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of $N = [a_{1}, a_{2}, ..., a_{k}]$ when $1⁄8$ is even. The $L(N) = [−a_{1}, a_{2} + 1, a_{3}, ..., a_{k}]$ are rational approximations to π and two successive terms always enclose the true value of π. Beginning with $R(N) = [a_{1}, 2, a_{2}, ..., a_{k}]$ the sequence starts ( / ):

 * $$ 2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385}, \frac{12465}{3968}, \frac{158720}{50521},\ldots \quad \longrightarrow \pi. $$

These rational numbers also appear in the last paragraph of Euler's paper cited above.

Consider the Akiyama–Tanigawa transform for the sequence ($N = [a_{1}, a_{2}, ..., a_{k}]$) /  ($±[a_{2}, ..., a_{k}]$):

 * {| class="wikitable" style="text-align:right;"

! 0 ! 1 ! 2 ! 3 ! 4 ! 5 ! 6
 * 1||$1⁄16$||0||−$1⁄2$||−$1⁄2$||−$3⁄8$||0
 * $1⁄4$|| 1|| $1⁄4$|| 0|| −$3⁄8$|| −$1⁄4$||
 * −1|| −$1⁄4$|| −$($a_{1}$)⁄($1⁄N!$)$|| $($σ_{n}(1)$)⁄($B_{1} = 1!(1⁄2!)$)$|| || ||
 * 8|| $1⁄2$|| || || || ||
 * }
 * }

From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −$1⁄2$ ×.

An algorithmic view: the Seidel triangle
The sequence Sn has another unexpected yet important property: The denominators of Sn+1 divide the factorial $B_{2} = 2!(−1⁄3! + 1⁄2!2!)$. In other words: the numbers $B_{3} = 3!(1⁄4! − 1⁄2!3! − 1⁄3!2! + 1⁄2!2!2!)$, sometimes called Euler zigzag numbers, are integers.

 * $$ T_n = 1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0, 1, 2, 3, \ldots $$ . See.

Their exponential generating function is the sum of the secant and tangent functions.

 * $$ \sum_{n=0}^\infty T_n \frac{x^n}{n!} = \tan \left(\frac\pi4 + \frac x2\right) = \sec x + \tan x$$.

Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as

 * $$\begin{align}

B_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [n\text{ even}] \frac{n }{2^n-4^n}\, T_{n-1}\ & n &\geq 2 \\ E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [n\text{ even}] T_{n} & n &\geq 0 \end{align}$$

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers $b(2n) = B_{2n}$ are given immediately by $n > 0$ and the Bernoulli numbers $b(3) = 3⁄2ζ(3)π^{−3}i$ are fractions obtained from $b(5) = −15⁄2ζ(5)π^{−5}i$ by some easy shifting, avoiding rational arithmetic.

What remains is to find a convenient way to compute the numbers $E_{2n}$. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate $2⁄π(4^{2n} − 2^{2n})$.

 * 1) Start by putting 1 in row 0 and let $B_{2n}$ denote the number of the row currently being filled
 * 2) If $B_{n} = E_{n} = 0$ is odd, then put the number on the left end of the row $B_{1}$ in the first position of the row $n ≥ 1$, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper
 * 3) At the end of the row duplicate the last number.
 * 4) If $sec x + tan x$ is even, proceed similar in the other direction.

Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont ) and was rediscovered several times thereafter.

Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers $S_{n}$ and recommended this method for computing $n$ and $n$ 'on electronic computers using only simple operations on integers'.

V. I. Arnold rediscovered Seidel's algorithm and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.

Triangular form:

 * {| style="text-align:right"

 * || || || || || || 1|| || || || || ||
 * || || || || || 1|| || 1|| || || || ||
 * || || || || 2|| || 2|| || 1|| || || ||
 * || || || 2|| || 4|| || 5|| || 5|| || ||
 * || || 16|| || 16|| || 14|| || 10|| || 5|| ||
 * || 16|| || 32|| || 46|| || 56|| || 61|| || 61||
 * 272|| ||272|| ||256|| ||224|| ||178|| ||122|| || 61
 * }
 * || || 16|| || 16|| || 14|| || 10|| || 5|| ||
 * || 16|| || 32|| || 46|| || 56|| || 61|| || 61||
 * 272|| ||272|| ||256|| ||224|| ||178|| ||122|| || 61
 * }
 * 272|| ||272|| ||256|| ||224|| ||178|| ||122|| || 61
 * }

Only, with one 1, and , with two 1s, are in the OEIS.

Distribution with a supplementary 1 and one 0 in the following rows:

 * {| style="text-align:right"

 * || || || || || || 1|| || || || || ||
 * || || || || || 0|| || 1|| || || || ||
 * || || || || −1|| || −1|| || 0|| || || ||
 * || || || 0|| || −1|| || −2|| || −2|| || ||
 * || || 5|| ||  5|| ||  4|| ||  2|| ||  0|| ||
 * || 0|| || 5|| || 10|| || 14|| || 16|| || 16||
 * −61|| ||−61|| ||−56|| ||−46|| ||−32|| ||−16|| || 0
 * }
 * || || 5|| ||  5|| ||  4|| ||  2|| ||  0|| ||
 * || 0|| || 5|| || 10|| || 14|| || 16|| || 16||
 * −61|| ||−61|| ||−56|| ||−46|| ||−32|| ||−16|| || 0
 * }
 * −61|| ||−61|| ||−56|| ||−46|| ||−32|| ||−16|| || 0
 * }

This is, a signed version of. The main andiagonal is. The main diagonal is. The central column is. Row sums: 1, 1, −2, −5, 16, 61.... See. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below.

The Akiyama–Tanigawa algorithm applied to ($R_{n} = 2S_{n}⁄S_{n + 1}$) / ($R_{n}$) yields:

 * {| style="text-align:right"

 * 1|| 1|| $N$|| 0|| −$N$|| −$N$|| −$n$
 * 0|| 1|| $ζ$|| 1|| 0|| −$i$
 * −1|| −1|| $n$|| 4|| $n$
 * 0|| −5|| −$n$|| 1
 * 5|| 5|| −$1⁄2$
 * 0|| 61
 * −61
 * }
 * 5|| 5|| −$1⁄4$
 * 0|| 61
 * −61
 * }
 * −61
 * }

1. The first column is. Its binomial transform leads to:

 * {| style="text-align:right"

 * 1|| 1|| 0|| −2|| 0|| 16|| 0
 * 0||−1||−2||2||16||−16
 * −1||−1||4||14||−32
 * 0||5||10||−46
 * 5||5||−56
 * 0||−61
 * −61
 * }
 * 5||5||−56
 * 0||−61
 * −61
 * }
 * −61
 * }
 * }

The first row of this array is. The absolute values of the increasing antidiagonals are. The sum of the antidiagonals is − ($n = 1$).

2. The second column is 1 1 −1 −5 5 61 −61 −1385 1385.... Its binomial transform yields:

 * {| style="text-align:right"

 * 1|| 2|| 2|| −4|| −16|| 32|| 272
 * 1||0||−6||−12||48||240
 * −1||−6||−6||60||192
 * −5||0||66||32
 * 5||66||66
 * 61||0
 * −61
 * }
 * 5||66||66
 * 61||0
 * −61
 * }
 * −61
 * }
 * }

The first row of this array is 1 2 2 −4 −16 32 272 544 −7936 15872 353792 −707584.... The absolute values of the second bisection are the double of the absolute values of the first bisection.

Consider the Akiyama-Tanigawa algorithm applied to ($n + 2$) / ( ($n + 1$) = abs( ($1⁄4$)) + 1 = 1, 2, 2, $1⁄8$, 1, $1⁄2$, $3⁄4$, $5⁄8$, 1, $3⁄4$, $1⁄2$, $1⁄2$....

 * {| style="text-align:right"

 * 1||2||2||$9⁄4$||1||$5⁄2$||$5⁄8$
 * −1||0||$7⁄2$||2||$3⁄4$||0
 * −1||−3||−$15⁄2$||3||$5⁄2$
 * 2||−3||−$11⁄2$||−13
 * 5||21||−$99⁄4$
 * −16||45
 * −61
 * }
 * 5||21||−$77⁄2$
 * −16||45
 * −61
 * }
 * −61
 * }

The first column whose the absolute values are could be the numerator of a trigonometric function.

is an autosequence of the first kind (the main diagonal is ). The corresponding array is:

 * {| style="text-align:right"

 * 0||−1||−1||2||5||−16||−61
 * −1||0||3||3||−21||−45
 * 1||3||0||−24||−24
 * 2||−3||−24||0
 * −5||−21||24
 * −16||45
 * −61
 * }
 * −5||−21||24
 * −16||45
 * −61
 * }
 * −61
 * }

The first two upper diagonals are −1 3 −24 402... = $n!$ ×. The sum of the antidiagonals is 0 −2 0 10... = 2 × (n + 1).

− is an autosequence of the second kind, like for instance /. Hence the array:

 * {| style="text-align:right"

 * 2||1||−1||−2||5||16||−61
 * −1||−2||−1||7||11||−77
 * −1||1||8||4||−88
 * 2||7||−4||−92
 * 5||−11||−88
 * −16||−77
 * −61
 * }
 * 5||−11||−88
 * −16||−77
 * −61
 * }
 * −61
 * }
 * }

The main diagonal, here 2 −2 8 −92..., is the double of the first upper one, here. The sum of the antidiagonals is 2 0 −4 0... = 2 × ($T_{n} = S_{n + 1} n!$1). − = 2 ×.

A combinatorial view: alternating permutations
Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis. Looking at the first terms of the Taylor expansion of the trigonometric functions $E_{2n}$ and $T_{2n}$ André made a startling discovery.

 * $$\begin{align}

\tan x &= x + \frac{2x^3}{3!} + \frac{16x^5}{5!} + \frac{272x^7}{7!} + \frac{7936x^9}{9!} + \cdots\\[6pt] \sec x &= 1 + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} + \frac{50521x^{10}}{10!} + \cdots \end{align}$$

The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of $B_{2n}$ has as coefficients the rational numbers $T_{2n - 1}$.

 * $$ \tan x + \sec x = 1 + x + \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 + \tfrac{5}{24}x^4 + \tfrac{2}{15}x^5 + \tfrac{61}{720}x^6 + \cdots $$

André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).

Related sequences
The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: $T_{n}$, $T_{n}$, $T_{n}$, $k$, $k$, / . Via the second row of its inverse Akiyama–Tanigawa transform, they lead to Balmer series  /.

The Akiyama–Tanigawa algorithm applied to ($k − 1$) /  ($61⁄2$) leads to the Bernoulli numbers  /,  / , or   without $k$, named intrinsic Bernoulli numbers $k$.

 * {| style="text-align:center; padding-left; padding-right: 2em;"

 * 1||$1⁄2$||$1⁄2$||$1⁄4$||$1⁄4$
 * 0||$1⁄8$||$3⁄2$||$3⁄4$||$3⁄2$
 * −$15⁄4$||−$15⁄2$||−$51⁄2$||−$n$||0
 * 0||−$3⁄2$||−$3⁄4$||−$3⁄4$||−$7⁄8$
 * }
 * 0||$17⁄16$||$17⁄16$||$33⁄32$||$3⁄2$
 * −$3⁄4$||−$3⁄4$||−$3⁄2$||−$5⁄4$||0
 * 0||−$3⁄2$||−$25⁄4$||−$27⁄2$||−$3⁄2$
 * }
 * 0||−$n$||−$5⁄6$||−$3⁄4$||−$7⁄10$
 * }

Hence another link between the intrinsic Bernoulli numbers and the Balmer series via ($T_{2n}$).

($B_{2n}$) = 0, 2, 1, 6,... is a permutation of the non-negative numbers.

The terms of the first row are f(n) = $E_{2n}$. 2, f(n) is an autosequence of the second kind. 3/2, f(n) leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5 ... = 1/2 + log 2.

Consider g(n) = 1/2 – 1 / (n+2) = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives:

 * {| style="text-align:center; padding-left; padding-right:2em;"

 * 0||$2⁄3$||$1⁄6$||$1⁄6$||$3⁄20$||$2⁄15$||...
 * 0||−$5⁄42$||−$1⁄30$||−$1⁄20$||−$2⁄35$||−$5⁄84$||...
 * $1⁄30$||$1⁄30$||$3⁄140$||$1⁄105$||0||−$1⁄42$||...
 * }
 * 0||−$1⁄28$||−$4⁄105$||−$1⁄28$||−$1⁄6$||−$1⁄4$||...
 * $3⁄10$||$1⁄3$||$5⁄14$||$1⁄6$||0||−$1⁄6$||...
 * }
 * $3⁄20$||$2⁄15$||$5⁄42$||$3⁄28$||0||−$1⁄30$||...
 * }

0, g(n), is an autosequence of the second kind.

Euler ($n + 1$) /  ($n$) without the second term ($1⁄20$) are the fractional intrinsic Euler numbers $n + 1$ The corresponding Akiyama transform is:

 * {| style="text-align:center; padding-left; padding-right: 2em;"

 * 1||1||$2⁄35$||$5⁄84$||$5⁄84$
 * 0||$1⁄30$||$1⁄30$||$3⁄140$||$1⁄105$
 * −$1⁄140$||−$1⁄2$||0||$7⁄8$||$3⁄4$
 * 0||−$21⁄32$||−$1⁄4$||−$3⁄8$||−$3⁄8$
 * }
 * −$5⁄16$||−$1⁄4$||0||$1⁄4$||$1⁄4$
 * 0||−$25⁄64$||−$1⁄2$||−$3⁄4$||−$9⁄16$
 * }
 * }
 * }
 * }

The first line is $n$. $n + 1$ preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are preceded by 0. The difference table is:

 * {| style="text-align:center; padding-left; padding-right: 2em;"

 * 0||1||1||$5⁄32$||$1⁄2$||$1⁄2$||$9⁄16$
 * 1||0||−$13⁄8$||−$125⁄64$||−$7⁄8$||−$3⁄4$||−$21⁄32$
 * −1||−$19⁄32$||0||$1⁄8$||$1⁄8$||$3⁄32$||$1⁄16$
 * }
 * −1||−$5⁄128$||0||$1⁄8$||$1⁄32$||$1⁄32$||$3⁄128$
 * }
 * }

Arithmetical properties of the Bernoulli numbers
The Bernoulli numbers can be expressed in terms of the Riemann zeta function as $(−1)^{n + 1}$ for integers $n +$ provided for $tan x$ the expression $sec x$ is understood as the limiting value and the convention $tan x + sec x$ is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that $1⁄64$ is a prime number if and only if $S_{n}$ is congruent to −1 modulo $p$. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

The Kummer theorems
The Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem, which says:

 * If the odd prime $p$ does not divide any of the numerators of the Bernoulli numbers $B_{0} = 1$ then $B_{1} = 0$ has no solutions in nonzero integers.

Prime numbers with this property are called regular primes. Another classical result of Kummer are the following congruences.

 * Let $p$ be an odd prime and $p$ an even number such that $B_{2} = 1⁄6$ does not divide $b$. Then for any non-negative integer $b$
 * $$ \frac{B_{k(p-1)+b}}{k(p-1)+b} \equiv \frac{B_{b}}{b} \pmod{p}. $$

A generalization of these congruences goes by the name of $B_{3} = 0$-adic continuity.

$B_{4} = −1⁄30$-adic continuity
If $k$, $b$ and $m$ are positive integers such that $n$ and $m$ are not divisible by $n + 4$ and $B_{1}$, then

 * $$(1-p^{m-1})\frac{B_m}{m} \equiv (1-p^{n-1})\frac{B_n} n \pmod{p^b}.$$

Since $B_{i}(n)$, this can also be written

 * $$\left(1-p^{-u}\right)\zeta(u) \equiv \left(1-p^{-v}\right)\zeta(v) \pmod{p^b},$$

where $n$ and $n − 2$, so that $n$ and $u$ are nonpositive and not congruent to 1 modulo $1⁄2 + 1⁄n + 2$. This tells us that the Riemann zeta function, with $n$ taken out of the Euler product formula, is continuous in the $v$-adic numbers on odd negative integers congruent modulo $n + 1$ to a particular $E_{i}(n) = 1, 0, −1⁄4, 0, 1⁄2, 0, −17⁄8, 0, ...$, and so can be extended to a continuous function $Eu(n)$ for all $p$-adic integers $$\mathbb{Z}_p,$$ the $p$-adic zeta function.

Ramanujan's congruences
The following relations, due to Ramanujan, provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition:

 * $$\binom{m+3}{m} B_m=\begin{cases}

\frac{m+3}{3}-\sum\limits_{j=1}^\frac{m}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 0\pmod 6;\\ \frac{m+3}{3}-\sum\limits_{j=1}^\frac{m-2}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 2\pmod 6;\\ -\frac{m+3}{6}-\sum\limits_{j=1}^\frac{m-4}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 4\pmod 6.\end{cases}$$

Von Staudt–Clausen theorem
The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt and Thomas Clausen independently in 1840. The theorem states that for every $Eu(n)$,
 * $$ B_{2n} + \sum_{(p-1)\,\mid\,2n} \frac1p$$

is an integer. The sum extends over all primes $B_{n} = −nζ(1 − n)$ for which $n ≥ 0$ divides $n = 0$.

A consequence of this is that the denominator of $−nζ(1 − n)$ is given by the product of all primes $B_{1} = 1⁄2$ for which $pB_{p − 1}$ divides $B_{2}, B_{4}, ..., B_{p − 3}$. In particular, these denominators are square-free and divisible by 6.

Why do the odd Bernoulli numbers vanish?
The sum

 * $$\varphi_k(n) = \sum_{i=0}^n i^k - \frac{n^k} 2$$

can be evaluated for negative values of the index $x^{p} + y^{p} + z^{p} = 0$. Doing so will show that it is an odd function for even values of $p − 1$, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that $p$ is 0 for $p$ even and $p − 1$; and that the term for $m ≡ n (mod p^{b − 1} (p − 1))$ is cancelled by the subtraction. The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for n > 1).

From the von Staudt–Clausen theorem it is known that for odd $B_{n} = −nζ(1 − n)$ the number $u = 1 − m$ is an integer. This seems trivial if one knows beforehand that the integer in question is zero. However, by applying Worpitzky's representation one gets

 * $$ 2B_n =\sum_{m=0}^n (-1)^m \frac{2}{m+1}m! \left\{{n+1\atop m+1} \right\} = 0\quad(n>1 \text{ is odd})$$

as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let $v = 1 − n$ be the number of surjective maps from $p − 1$ to $1 − p^{−s}$, then $p − 1$. The last equation can only hold if

 * $$ \sum_{\text{odd }m=1}^{n-1} \frac 2 {m^2}S_{n,m}=\sum_{\text{even } m=2}^n \frac{2}{m^2} S_{n,m} \quad (n>2 \text{ is even}). $$

This equation can be proved by induction. The first two examples of this equation are

Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.

A restatement of the Riemann hypothesis
The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the Riemann hypothesis (RH) which uses only the Bernoulli numbers. In fact Marcel Riesz proved that the RH is equivalent to the following assertion:

 * For every $a ≢ 1 mod (p − 1)$ there exists a constant $ζ_{p}(s)$ (depending on $n > 0$) such that $p$ as $p − 1$.

Here $2n$ is the Riesz function

 * $$ R(x) = 2 \sum_{k=1}^\infty

\frac{k^{\overline{k}} x^{k}}{(2\pi)^{2k}\left(\frac{B_{2k}}{2k}\right)} = 2\sum_{k=1}^\infty \frac{k^{\overline{k}}x^k}{(2\pi)^{2k}\beta_{2k}}. $$

$B_{2n}$ denotes the rising factorial power in the notation of D. E. Knuth. The numbers $p$ occur frequently in the study of the zeta function and are significant because $p − 1$ is a $2n$-integer for primes $n$ where $k$ does not divide $B_{2k + 1 − m}$. The $m$ are called divided Bernoulli numbers.

Generalized Bernoulli numbers
The generalized Bernoulli numbers are certain algebraic numbers, defined similarly to the Bernoulli numbers, that are related to special values of Dirichlet $p$-functions in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.

Let $L$ be a Dirichlet character modulo $χ$. The generalized Bernoulli numbers attached to $f$ are defined by

 * $$\sum_{a=1}^f \chi(a) \frac{te^{at}}{e^{ft}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}.$$

Apart from the exceptional $2k + 1 − m > 1$, we have, for any Dirichlet character $χ$, that $B_{1}$ if $n > 1$.

Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers $2B_{n}$:

 * $$L(1-k,\chi)=-\frac{B_{k,\chi}}k,$$

where $S_{n,m}$ is the Dirichlet $χ$-function of $L$.

Eisenstein–Kronecker number
Eisenstein–Kronecker numbers are an analogue of the generalized Bernoulli numbers for imaginary quadratic fields. They are related to critical L-values of Hecke characters.

title: Bernoulli number