Anna Kiss

Interweaving

Tamás Varga

Interweaving probability games and other mathematical areas in Tamás

Keywords: Complex mathematics education, probability, games

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Short description of the research topic

The aim of the presentation is to show how it was possible in a pilot project to interweave probability and other mathematical areas in the spirit of Tamás Varga's complex mathematics education model in Hungary.

The project was launched under the Recent Complex Mathematics Education Project1 , the aim of which is to examine how Tamás Varga's relatively old, but no way outdated ideas can be implemented to match present-day challenges effectively. In this sense it is also an interesting question, I intend to address in my later research, at what point digital devices should be introduced to help discovery. As Tamás Varga proposed to use manipulatives for the earliest stages of discovery, in this presentation two games accompanied by using manipulatives are presented.

"Complex" in this sense stands for teaching different areas of mathematics at the same time, often as problems of mathematical nature arise in the real world, and thus giving students a more complex and realistic image of what mathematics and mathematical thinking is.

Theoretical framework

Even though Tamás Varga introduced probability from early primary school [START_REF] Varga | Logic and Probability in the Lower Grades[END_REF] in his curriculum as early as in the 1960s [START_REF] Varga | A matematika tanítása[END_REF], in Hungary, similarly to many other countries (Serradó, Azcárate, & Cardenoso, 2006) the majority of teachers tried to avoid or minimize teaching probability in their teaching practice [START_REF] Pálfalvi | Matematika didaktikusan [Mathematics didactically[END_REF]. In today's curricula there are three lessons appointed in fifth grade for dealing with "possible, impossible and sure" situations. In sixth grade, no probability is included, but a chapter for statistics at the very end of the schoolyear, where in practice it is often omitted.

"Guided experience with randomness in earlier years is an important prerequisite to successful teaching of formal probability" (Steen, 1990, p. 120.). In this study guided experience was embedded into the lessons appointed for teaching fractions and arithmetic. This way, instead of linear mathematics teaching, the teacher engaged in complex mathematics teaching where consolidating other mathematical areas and guided experience with randomness became possible at the same time. This way probabilistic concepts could be introduced without need for more lessons. This research would be worth to be continued to enable teachers to teach more mathematical areas simultaneously.

The method

In the project, probability games were selected to be taught in a complex way in four consecutive schoolyears for fifth and sixth graders, two of which are shown in the poster. One, a modified version of the game called "pig", where 5 and 6 are out ruled (Tsao, 2016) connects probability and fractions [START_REF] Sinicrope | Multiplication of Fractions through Paper Folding[END_REF], the other, played with two tetrahedron dice [START_REF] Nilsson | Different ways in which students handle chance encounters in the explorative setting of a dice game[END_REF] connects probability and arithmetic. Student-teacher dialogues were transcribed to be analyzed in a later article to further investigate students' knowledge construction processes [START_REF] Ron | Partially correct constructs illuminate students' inconsistent answers[END_REF].

Research results

The aim of the pilot project was to introduce probabilistic thinking into lessons where other mathematical areas were in the focus through complex teaching. We have found that through careful planning and selection of games, it was possible to teach more in fewer lessons, and thus "engage children in considerations of chance more often and more systematically" (Lehrer, & English, 2018, p. 248).

This paper was supported by the Hungarian Academy of Sciences under the Recent Complex Mathematics Education Project


Alin Bostan
email: alin.bostan@inria.fr

Jordan Tirrell
email: jtirrell@mtholyoke.edu

Bruce W Westbury
email: bruce.westbury@utdallas.edu.§zhangy@amss.ac.cn.

Yi Zhang

On sequences associated to the invariant theory of rank two simple Lie algebras

Keywords: representation theory, Lie algebra, binomial transform, algebraic residues, computer algebra, creative telescoping

We study two families of sequences, listed in the On-Line Encyclopedia of Integer Sequences (OEIS), which are associated to invariant theory of Lie algebras. For the first family, we prove combinatorially that the sequences A059710 and A108307 are related by a binomial transform. Based on this, we present two independent proofs of a recurrence equation for A059710, which was conjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs' conjecture by the method of algebraic residues. As a consequence, closed formulae for the generating function of sequence A059710 are obtained in terms of classical Gaussian hypergeometric functions. Moreover, we show that sequences in the second family are also related by binomial transforms.

Introduction

The representation theory of simple Lie algebras is a keystone of algebraic and enumerative combinatorics, giving rise to combinatorial objects such as tableaux, symmetric functions, quantum groups, crystal graphs, and many others. We are interested in two families of sequences in OEIS. Sequences in the first family of sequences are called octant sequences. For instance, sequence A059710, which is the first octant sequence, is defined to be a sequence associated to fundamental representations of the exceptional simple Lie algebra G 2 , of rank two and dimension fourteen [START_REF] Kuperberg | Spiders for rank 2 Lie algebras[END_REF]. The second octant sequence is A108307 [START_REF] Lin | Restricted inversion sequences and enhanced 3-noncrossing partitions[END_REF], and it is defined to be the cardinality of the set of set partitions of [n] with no enhanced 3-crossing. Our first contribution is to prove that sequences A059710 and A108307 are tightly related, by a binomial transform.

Theorem 1.1. Let T 3 (n) and E 3 (n) be the n-th terms of A059710 and A108307, respectively. Then E 3 is the binomial transform of T 3 , i.e., for n ≥ 0,

E 3 (n) = n ∑ k=0 n k T 3 (k).
Theorem 1.1 provides an unexpected connection between invariant theory of G 2 and combinatorics of set partitions. In the same spirit, [9, Theorem 1] proves a binomial relation between E 3 and A108304, which is the third octant sequence. Together, these two results show that the octant sequences are associated to representations of G 2 .

Building on Theorem 1.1, as well as on results by Bousquet-Mélou and Xin [START_REF] Bousquet | On partitions avoiding 3-crossings[END_REF], our second contribution is to give two independent proofs of a recurrence equation for T 3 conjectured by Mihailovs [18, §3], which was the initial motivation for our study: Theorem 1.2. The sequence T 3 is determined by the initial conditions T 3 (0) = 1, T 3 (1) = 0, T 3 (2) = 1 and the recurrence relation that for n ≥ 0, 14

(n + 1) (n + 2) T 3 (n) + (n + 2) (19n + 75) T 3 (n + 1) + 2 (n + 2) (2n + 11) T 3 (n + 2) -(n + 8) (n + 9) T 3 (n + 3) = 0. (1.1)
Furthermore, we give an alternative proof of Theorem 1.2, using the interpretation of T 3 in terms of G 2 walks and using algorithms for computing Picard-Fuchs differential equations for algebraic residues. As a consequence, closed formulae for the generating function of T 3 are obtained in terms of the classical Gaussian hypergeometric function.

We consider a second family of sequences, called quadrant sequences. These are defined to be sequences associated to representations of G 2 restricted to SL [START_REF] Bostan | Supplementary electronic material to the article[END_REF]. By invariant theory, these sequences are also are related by binomial transforms. Based on this, we give a uniform recurrence equation holding for each quadrant sequence. Furthermore, we show that sequences in the second family are identical to quadrant sequences because they satisfy the same initial conditions and recurrence equations. They are related to the octant sequences by the branching rules [START_REF] Gaskell | Generating functions for G 2 characters and subgroup branching rules[END_REF] for the maximal subgroup SL(3) of G 2 .

Preliminaries

In this section, we give a general construction of sequences associated to the invariant theory of algebraic groups and the relations between them. Definition 2.1. Let G be a reductive complex algebraic group and let V be a (finite dimensional) representation of G. The sequence associated to (G, V), denoted a V , is the sequence whose n-th term is the multiplicity of the trivial representation in the tensor power ⊗ n V.

Given the sequence a with n-th term a(n), the binomial transform of a is the sequence, denoted Ba, whose n-th term is

n ∑ i=0 n i a(i).
Here, B denotes the binomial transform operator. Binomial transforms arise naturally for sequences associated to the representations of reductive complex algebraic groups.

Lemma 2.2. Assume that a V is the sequence associated to (G, V) as specified in Definition 2.1.

Then a V⊕C = Ba V .

The binomial transform also arises naturally for lattice walks restricted to a domain.

Lemma 2.3. Assume that a sequence a enumerates walks in a lattice, confined to a domain D, using a set of steps S. Then Ba is given by adding a new step corresponding to the zero element of the lattice; that is, S is replaced by the disjoint union S ∐{0} without changing the domain.

The binomial transform can also be regarded as an operator on the generating function of a sequence. Let G(t) = ∑ ∞ n=0 a(n)t n be the generating function of the sequence a. We denote the generating function of B k a by B k G. Then we have Lemma 2.4. For k ∈ Z, the k-th binomial transform of G(t) is

(B k G)(t) = 1 1 -k t G t 1 -k t .

Octant sequences

The first family of sequences we are interested in is illustrated in Figure 1. In Subsection 3.1, we show that A059710 and A108307 are related by the binomial transform, which is Theorem 1.1. Combining with [9, Theorem 1] and Lemma 2.2, sequences in this first family are identical to the sequences a V⊕k C (Definition 2.1), where V is the seven dimensional fundamental representation of the exceptional simple Lie algebra G 2 , for k = 0, 1, 2. They can also be interpreted in terms of lattice walks restricted to the octant. Thus, we call them octant sequences. Let T 3 (n) be the n-th term of the first octant sequence A059710. In Subsection 3.2, based on Theorem 1.1, we present two independent proofs of Theorem 1.2, which gives a linear recurrence equation for T 3 (n). Besides, we also give a direct proof of Theorem 1.2 by the method of algebraic residues. As a consequence, closed formulae for the generating function of T 3 are presented in terms of hypergeometric functions in Subsection 3.

Lattice walks

In this subsection we prove Theorem 1.1, using lattice walks. An excursion is a walk which starts and ends at the origin. The main idea of our proof is to exhibit excursions that are enumerated by the two sequences and then to compare these.

Sequence T 3 has an interpretation via enumerating excursions on the weight lattice of G 2 . This interpretation is given in [START_REF] Kuperberg | Spiders for rank 2 Lie algebras[END_REF][START_REF] Westbury | Enumeration of non-positive planar trivalent graphs[END_REF] and it uses the theory of Kashiwara crystals. Proposition 3.1. Let C be the crystal of the representation V. For n 0, T 3 (n) is the number of highest weight words of weight 0 in the tensor power ⊗ n C.

The highest weight words are in bijection with a set of excursions in the weight lattice of G 2 . The steps are the seven weights of the fundamental representation V. The walk is constrained to stay in the dominant chamber. These weights and the dominant chamber are shown in Figure 2. There is one further rule, which is the boundary rule, stating that if a walk is at a boundary point (0, y), then the step (0, 0) is not permitted.

Let E 3 (n) be the n-th term of the second octant sequence A108307. Then the se- quence E 3 enumerates hesitating tableaux of height 2, introduced in [START_REF] William | Crossings and nestings of matchings and partitions[END_REF]. The excursion interpretation is given in [START_REF] Bousquet | On partitions avoiding 3-crossings[END_REF].

A hesitating tableau of semilength n is an excursion in the Young lattice with n steps. Each step is one of the following pairs of moves on the Young lattice:

• do nothing, add a cell;

• remove a cell, do nothing;

• add a cell, remove a cell.

A hesitating tableau of height h is a hesitating tableau such that every partition in the sequence has height at most h. A hesitating tableau of height 2 can be interpreted as an excursion in Z 2 by identifying partitions with at most two nonzero rows with the set {(x, y) ∈ Z 2 | x y 0.} There are eight steps since there are two ways to do nothing then add a cell, two ways to remove a cell then do nothing and four ways to remove a cell then add a cell. Two

(1,0) (-1,1) (-2,1) (-1,0) (1,-1)
(2,-1) (0,0) (1,0)

(0,1) (-1,1) (-1,0) (0,-1) (1,-1)
(0,0) of the four ways to remove a cell then add a cell give the step (0, 0), namely add a cell on the first row then remove this cell and add a cell on the second row then remove this cell. It is always allowed to add and then remove a cell on the first row. The step which adds and removes a cell on the second row is not permitted on the line x = y.

Proof of Theorem 1.1. We compare the two descriptions of the walks. The steps of the walks for the sequence T 3 are shown in Figure 2 and the steps of the walks for the sequence E 3 are shown in Figure 3.

In order to compare these, we first make the change of coordinates (x, y) → (x, y ± x). This identifies the six non-zero steps, as well as the two domains.

By Lemma 2.3, it remains to compare the zero steps. There is one zero step in the T 3 case and two in the E 3 case. In the E 3 case we have the zero step which adds and then removes a cell on the second row. The boundary condition is that this step is not allowed on the line x = y. After the change of coordinates, this is the same boundary condition as the zero step in the T 3 case. The second zero step in the E 3 case adds and then removes a cell in the first row. This is always allowed.

Algebra

In this subsection, we give three proofs of Theorem 1.2, with different flavors. The first two proofs use Theorem 1.1 and a result of Bousquet-Mélou and Xin [START_REF] Bousquet | On partitions avoiding 3-crossings[END_REF] on partitions that avoid 3-crossings. The last proof relies on the connection with G 2 walks.

First proof of Theorem 1.2

The first proof is based on Theorem 1.1, on Proposition 2 in [START_REF] Bousquet | On partitions avoiding 3-crossings[END_REF] and on the method of creative telescoping [START_REF] Zeilberger | The method of creative telescoping[END_REF][START_REF] Koutschan | Advanced applications of the holonomic systems approach[END_REF] for the summation of (bivariate) holonomic sequences. Proposition 3.2. [4, Proposition 2] The number E 3 (n) of partitions of [n] having no enhanced 3-crossing is given by E 3 (0) = E 3 (1) = 1, and for n ≥ 0,

8 (n + 3) (n + 1) E 3 (n) + 7n 2 + 53n + 88 E 3 (n + 1) -(n + 8) (n + 7) E 3 (n + 2) = 0. (3.1)
Equivalently, the associated generating function

E (t) = ∑ n≥0 E 3 (n)t n satisfies t 2 (1 + t) (1 -8t) d 2 dt 2 E (t) + 2t 6 -23t -20t 2 d dt E (t) + 6 5 -7t -4t 2 E (t) -30 = 0.
First Proof of Theorem 1.2. By Theorem 1.1, we have

T 3 (n) = n ∑ k=0 (-1) n-k n k E 3 (k). Set f (n, k) = (-1) n-k ( n k )E 3 (k)
. By Proposition 3.2, and by the closure properties for holonomic functions, it follows that f (n, k) is holonomic. Thus, we can apply Chyzak's algorithm [START_REF] Koutschan | Advanced applications of the holonomic systems approach[END_REF] for creative telescoping to derive a recurrence relation for T 3 . In particular, using the Koutschan's Mathematica package HolonomicFunctions.m [START_REF] Koutschan | HolonomicFunctions user's guide[END_REF] that implements Chyzak's algorithm, we find exactly the recurrence equation in Theorem 1.2.

The detailed calculation involved in the above proof can be found in [START_REF] Bostan | Supplementary electronic material to the article[END_REF].

Second proof of Theorem 1.2

The second proof is also based on Proposition 2 in [START_REF] Bousquet | On partitions avoiding 3-crossings[END_REF] and on Theorem 1.1, namely on the relation between the generating functions of T 3 (n) and of E 3 (n) implied by Theorem 1.1.

Second Proof of Theorem 1.2. Let T (t) = ∑ n≥0 T 3 (n)t n . By Theorem 1.1 and Lemma 2.4,

T (t) = 1 1 + t • E t 1 + t .
We know from Proposition 3.2 a differential equation for E (t). By (univariate) closure properties of D-finite functions, we deduce a differential equation for T (t), and convert it into a linear recurrence for T 3 (n), which is exactly the recurrence in Theorem 1.2.

Third proof of Theorem 1.2

The third proof of Theorem 1.2 relies on G 2 walks and the method in [START_REF] Bostan | Hypergeometric expressions for generating functions of walks with small steps in the quarter plane[END_REF]. First we define two elements K and W in the group ring of the root lattice of G 2 . Since the root lattice has rank 2, these become Laurent polynomials in two variables once we choose a basis of the root lattice. The element K is the character of the representation V; W is given by

W = ∑ w∈W ε(w) [w(ρ) -ρ]
Here W is the Weyl group, ε : W → {±1} is the sign character and ρ is half the sum of the positive roots.

The following definition is given in [START_REF] Kuperberg | The quantum G 2 link invariant[END_REF] in the paragraph following Conjecture 3.3. The general principle is discussed in [START_REF] Gessel | Random walk in a Weyl chamber[END_REF].

Definition 3.3. The n-th term T 3 (n) is the constant term of the Laurent polynomial W K n , where

K = (1 + x + y + x y + x -1 + y -1 + (xy) -1 ),
and W is the Laurent polynomial

W = x -2 y -3 (x 2 y 3 -xy 3 + x -1 y 2 -x -2 y + x -3 y -1 -x -3 y -2 + x -2 y -3 -x -1 y -3 + xy -2 -x 2 y -1 + x 3 y -x 3 y 2 ). Third Proof of Theorem 1.2. Let T (t) = ∑ n≥0 T 3 (n)t n . By Definition 3.3, T (t) is the con- stant coefficient [x 0 y 0 ] of W/(1 -tK).
In other words, T (t) is equal to the algebraic residue of W/(xy -txyK), and thus it is D-finite [START_REF] Lipshitz | The diagonal of a D-finite power series is D-finite[END_REF] and is annihilated by a linear differential operator that cancels the contour integral of W/(xy -txyK) over a cycle. Using the integration algorithm for multivariate rational functions in [START_REF] Bostan | Creative telescoping for rational functions using the Griffiths-Dwork method[END_REF] we compute the following operator of order 6, that cancels T (t):

L 6 = t 5 (t + 1) (7 t -1) (2 t + 1) 2 ∂ 6 + 3 t 4 (2 t +
Q = (2 t + 1) t 3 ∂ 3 + (24 t + 13) t 2 ∂ 2 + 6 (12 t + 7) t∂ + 48 t + 30,
and

L 3 = t 2 (2 t + 1) (7 t -1) (t + 1) ∂ 3 + 2 t (t + 1) 63 t 2 + 22 t -7 ∂ 2 + 252 t 3 + 338 t 2 + 36 t -42 ∂ + 28 t (3 t + 4) .
This shows that f (t) := L 3 (T (t)) is a solution of the differential operator Q. Hence, by denoting f (t) = ∑ n≥0 f n t n , one deduces that for all n ≥ 0 we have

2 (n + 2) f n + (n + 6) f n+1 = 0.
One the other hand, from T (t) = 1 + t 2 + O(t 3 ), it follows that f 0 = 0, therefore f (t) = 0, in other words T (t) is also a solution of L 3 . From there, deducing the recurrence relation of Theorem 1.2 is immediate.

Closed formulae

By factorization of the operator L 3 and using algorithms for solving second order differential equations, as described in [START_REF] Bostan | Hypergeometric expressions for generating functions of walks with small steps in the quarter plane[END_REF], we derive the following closed formula for T (t):

T (t) = 1 30 t 5 R 1 • 2 F 1 1 3 2 3 2 ; φ + R 2 • 2 F 1 2 3 4 3 3 ; φ + 5 P ,
where Following the approach in [1, §3.3], itself based on [17, Theorem 1.5], one can obtain an alternative expression using a more geometric flavor. The key point is that the denominator of W/(xy -txyK) is a family of elliptic curves, thus integrating W/(xy -txyK) over a small torus amounts to computing the periods of the two forms (of the first and second kind). Working out the details, this approach yields an expression for T (t) in terms of the Weierstrass invariant g 2 = (t -1) 25 t 3 + 21 t 2 + 3 t -1 and of the j-invariant J = (t -1) 3 25 t 3 + 21 t 2 + 3 t -1 3 t 6 (1 -7 t) (2 t + 1) 2 (t + 1) 3 of our family of curves. As in [START_REF] Bostan | Hypergeometric expressions for generating functions of walks with small steps in the quarter plane[END_REF], we introduce the expression

R 1 = (t +
H(t) = 1 g 2 1/4

Quadrant sequences

The second family of sequences we are interested in is given in Figure 4. We show that sequences in this second family are identical to sequences associated to representations of G 2 restricted to SL(3). Thus, by Lemma 2.2, those sequences are also related by binomial transforms. Moreover, they can also be interpreted as lattice walks restricted to the quadrant. This is why we call them quadrant sequences. They are related to the octant sequences by the branching rules [START_REF] Gaskell | Generating functions for G 2 characters and subgroup branching rules[END_REF] for the maximal subgroup SL(3) of G 2 . and

W = 1 - x 2 y + x 3 -x 2 y 2 + y 3 - y 2 x . ( 4.2)
Let C 2 (n) be the n-th term of the sequence A216947 in the second family. Marberg [16, Theorem 1.7] showed C 2 (n) is the constant term [x 0 y 0 ] of W Kn , where K = K| k=3 , the Laurent polynomials K and W are specified in (4.1) and (4.2). By Lemma 4.2, the sequence S 3 is identical to the sequence A216947. Therefore, we have: Proposition 4.3. [16, Theorem 1.7] The n-th term C 2 (n) of the sequence S 3 is given by C 2 (0) = 1, C 2 (1) = 3 and for n ≥ 0:

(n + 5)(n + 6) • C 2 (n + 2) -2(5n 2 + 36n + 61) • C 2 (n + 1) + 9(n + 1)(n + 4) • C 2 (n) = 0,
Equivalently, the associated generating function

C(t) = ∑ n≥0 C 2 (n)t n satisfies 72C(t) + 4(-61 + 117t) d dt C(t) + 2(15 -184t + 234t 2 ) d 2 dt 2 C(t) + 2t(-6 + 7t)(-1 + 9t) d 3 dt 3 C(t) + (-1 + t)t 2 (-1 + 9t) d 4 dt 4 C(t) = 0.
Next, for k = 0, 1, 2, 3, we prove a uniform recurrence equation for the sequence S k . It is given by a single formula with k as a parameter. Moreover, we show that S 0 , S 1 , S 2 , S 3 are identical to sequences in the second family. Theorem 4.4. For k = 0, 1, 2, 3, the n-th term a(n) of the sequence S k satisfies the following recurrence equation: In [START_REF] Marberg | Crossings and nestings in colored set partitions[END_REF], Marberg proved that A151366, A001181, and A216947 are related by binomial transforms by using combinatorial methods. Here, we give another proof by the representation theory of simple Lie algebras. Proof. In (4.3), by setting k to 0, 1, 2, 3, we find that the corresponding recurrence operators are left multiples [5, page 618] of those of A216947, A001181, A236408, and A151366 specified in OEIS. It implies that S 0 , S 1 , S 2 , S 3 satisfy the same recurrence equations as the sequences in the second family. To verify they are the same sequences, we just need to check finitely many initial terms. The details of the verification can be find in [START_REF] Bostan | Supplementary electronic material to the article[END_REF]. Since S k 's are related by binomial transforms, so are sequences in the second family.

(k -9)(k -1)k 2 (n + 1)(n + 2)a(n)+ 2k(n + 2)(2k 2 n -15kn + 8k 2 + 9n -56k + 36)a(n + 1)+ (6k 2 n 2 + 54k 2 n -30kn 2 + 114k 2 + 9n 2 -254kn + 81n -510k + 162)a(n + 2) + 2(2kn

Figure 1 :

 1 Figure 1: The first family of sequences a, with their OEIS tags, and their first terms.

Figure 2 :

 2 Figure 2: Steps in weight lattice

Figure 3 :

 3 Figure 3: Steps in octant

Definition 4 . 1 .Lemma 4 . 2 .

 4142 Let V be the defining representation of SL(3) and denote the dual by V * . For k 0, the quadrant sequences S k are the sequence associated to (SL(3), V ⊕ V * ⊕ k C). Let C k be the generating function of S k , where k ≥ 0. Then C k is the constant coefficient of [x 0 y 0 ] of W/(1 -tK), whereK = k + x + y + x -1

Figure 4 :

 4 Figure 4: The second family of sequences a, their OEIS tags and their first terms.

Corollary 4 . 5 .

 45 The sequences S 0 , S 1 , S 2 , S 3 are identical to sequences in the second family specified in Figure4. Moreover, sequences in the second family are related by binomial transforms.

  1) 168 t 3 + 211 t 2 + 40 t -11 ∂ 5 + 6 t 3 2100 t 4 + 3475 t 3 + 1616 t 2 + 79 t -61 ∂ 4 + 6 t 2 11200 t 4 + 17400 t 3 + 7556 t 2 + 268 t -273 ∂ 3 + 36 t 4200 t 4 + 6100 t 3 + 2442 t 2 + 54 t -77 ∂ 2 + 36 3360 t 4 + 4540 t 3 + 1646 t 2 + 16 t -35 ∂+ 20160 t 3 + 25200 t 2 + 8064 t, where ∂ = ∂ ∂t denotes the derivation operator with respect to t. The operator L 6 factors as L 6 = QL 3 , where

  1) 2 214 t 3 + 45 t 2 + 60 t + 5 t -1 , R 2 = 6 t 2 (t + 1) 2 101 t 2 + 74 t + 5 P = 28 t 4 + 66 t 3 + 46 t 2 + 15 t + 1.

	and	φ =	27 (t + 1) t 2 (1 -t) 3 ,
			(t -1) 2	,

  • 2 F 1 + 182 t + 59 (11 t + 1) H (t) + 341 t 3 + 507 t 2 + 231 t + 1 (5 t + 1) H ′ (t) .

			1 12	1	5 12	;	1728 J	.
	Then T (t) is proved to be equal to
	P 6 t 5 +	(7 t -1) (2 t + 1) (t + 1) 360 t 5	155 t 2

  2 + 24kn -5n 2 + 70k -56n -153)a(n + 3) + (n + 7)(n + 8)a(n + 4) = 0 (4.3) Proof. By Lemma 2.2, sequences S 2 , S 1 , S 0 are the first, second, and third binomial transforms of S 3 , respectively. Thus, by Lemma 2.4, the generating function of S k is C(t) is the generating function of S 3 . Regarding k as a parameter in the above expression, we deduce the differential equation for A(t) in the claim by using Proposition 4.3 and closure properties of D-finite functions. By converting the differential equation for A(t), we get the corresponding recurrence equation for the sequence a(n), which is exactly the recurrence equation in the claim.

	A(t) :=	1 1 -kt	• C	t 1 -kt	for k = 0, 1, 2, 3,

where

Yi Zhang was partially supported by the UTD Program P-1-03246, the Natural Science Foundation of USA Grants CCF-1815108 and CCF-1708884.