Delannoy Numbers Research Articles

Alternative proofs of some formulas for two tridiagonal determinants

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

Some congruences on Delannoy numbers and Schröder numbers

The Delannoy numbers and Schröder numbers are given by [Formula: see text] respectively. We mainly prove that for any prime [Formula: see text], [Formula: see text] which was originally conjectured by Sun in 2011. A related congruence on the Delannoy numbers is also proved.

An overpartition analogue of q-binomial coefficients, II: Combinatorial proofs and (q,t)-log concavity

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an m×n rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization [m+nn]‾q,t of Gaussian polynomials, which is also a (q,t)-analogue of Delannoy numbers. First we obtain finite versions of classical q-series identities such as the q-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite theta function and prove the (q,t)-log concavity of [m+nn]‾q,t. We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of [m+nn]‾q,t.

Positivity of Iterated Sequences of Polynomials

In this paper, we present some criteria for the 2-$q$-log-convexity and 3-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of a certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$ Those criteria can also be presented by continued fractions and generating functions. These allow a unified treatment of the 2-$q$-log-convexity of alternating Eulerian polynomials, 2-log-convexity of Euler numbers, and 3-$q$-log-convexity of many classical polynomials, including the Bell polynomials, the Eulerian polynomials of types $A$ and $B$, the $q$-Schröder numbers, $q$-central Delannoy numbers, the Narayana polynomials of types $A$ and $B$, the generating functions of rows in the Catalan triangles of Aigner and Shapiro, the generating functions of rows in the large Schröder triangle, and so on, which extend many known results for $q$-log-convexity.

Arithmetic properties of Delannoy numbers and Schröder numbers

DefineDn(x)=∑k=0n(nk)2xk(x+1)n−kforn=0,1,2,… andsn(x)=∑k=1n1n(nk)(nk−1)xk−1(x+1)n−kforn=1,2,3,…. Then Dn(1) is the n-th central Delannoy number Dn, and sn(1) is the n-th little Schröder number sn. In this paper we obtain some surprising arithmetic properties of Dn(x) and sn(x). We show that1n∑k=0n−1Dk(x)sk+1(x)∈Z[x(x+1)]for alln=1,2,3,…. Moreover, for any odd prime p and p-adic integer x≢0,−1(modp), we establish the supercongruence∑k=0p−1Dk(x)sk+1(x)≡0(modp2). As an application we confirm Conjecture 5.5 in [S14a], in particular we prove that1n∑k=0n−1TkMk(−3)n−1−k∈Zfor alln=1,2,3,…, where Tk is the k-th central trinomial coefficient and Mk is the k-th Motzkin number.

Some properties of central Delannoy numbers

In the paper, by investigating the generating function of central Delannoy numbers, the authors establish several explicit expressions, including determinantal expressions, for central Delannoy numbers, present three identities involving the Cauchy products of central Delannoy numbers, discover an integral representation for central Delannoy numbers, find (absolute) monotonicity, convexity, and logarithmic convexity for the sequence of central Delannoy numbers, and construct several product and determinantal inequalities for central Delannoy numbers.

Some properties of the Schröder numbers

In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schroder numbers and find three relations between the Schroder numbers and central Delannoy numbers. Moreover, the authors sketch generalizing the Schroder numbers and central Delannoy numbers and their generating functions.

Powers of a Class of Generating Functions

SummaryWe consider positive powers of a rational generating function F(x, z) of the variable z whose numerator has degree at most 1, denominator has degree 2, and where coefficients are functions of x. Many well-known polynomial sequences are generated by functions of this form, including the generating function U(x, z) for Chebyshev polynomials Un(x) of the second kind; the square root U(x, z)1/2 generates the sequence of Legendre polynomials. The numerical sequence Un(3) is the sequence of numbers whose squares are triangular numbers, and the square root of its generating function breeds the sequence of central Delannoy numbers. Our goal is to provide a recurrence for the sequence generated by F(x, z)α, where α is a positive real number, thus providing a reasonable way for computing sequential values.

A generalized neighborhood for cellular automata

A generalized neighborhood for d-dimensional cellular automata is introduced; it spans the range from von Neumann's to Moore's neighborhood using a parameter which represents the dimension of hypercubes connecting neighboring cells. Finite hypercubes and hypertoruses are studied, and the number of neighbors on their boundary and the number of connections between cells are calculated. We come to finite constructs when practically implementing computations of cellular automata. Enumerations of cells, neighbors and connections are considered and implemented in ad-hoc software which generates a canvas of hypercube and hypertorus models in the form of a Petri net. A cell model can be replaced while the underlying canvas of connections remains the same. The generalized neighborhood is extended to include a concept of radius; the number of neighbors is calculated for infinite and finite lattices. For diamond-shaped neighborhoods, a sequence is obtained whose partial sums equal Delannoy numbers.

The reciprocal of the weighted geometric mean is a Stieltjes function

In the paper, by the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the reciprocal of the weighted geometric mean of two positive numbers. By the integral representation, the authors derive that the reciprocal of the weighted geometric mean of two positive numbers is a Stieltjes function and consequently a (logarithmically) completely monotonic function, present several integral formulas for some kinds of improper integrals, deduce several integral representations for some functions related to the weighted geometric mean, find an equivalent relation between integral representations for the weighted geometric mean and its reciprocal, connect the integral formulas with the central Delannoy numbers, and apply some integral representations and some integral formulas to compute some concrete improper integrals appeared in the theory of complex functions.

Proof of Sun's conjectures on integer-valued polynomials

Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to prove that certain rational sums involving even powers of those polynomials are integers whenever they are evaluated at integers. This confirms two conjectures of Z.-W. Sun. We also conjecture that many of these results have neat q-analogues.

A supercongruence involving Delannoy numbers and Schröder numbers

The Delannoy numbers and Schröder numbers are given byDn=∑k=0n(nk)(n+kk)andSn=∑k=0n(nk)(n+kk)1k+1, respectively. Let p>3 be a prime. We mainly prove that∑k=1p−1DkSk≡2p3Bp−3−2pHp−1⁎(modp4), where Bn is the n-th Bernoulli number and these Hn⁎ are the alternating harmonic numbers given by Hn⁎=∑k=1n(−1)kk. This supercongruence was originally conjectured by Z.-W. Sun in 2011.

Catalan-like numbers and Stieltjes moment sequences

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schröder numbers, are Stieltjes moment sequences in a unified approach.

Some congruences related to the q-Fermat quotients

We give q-analogues of the following congruences by Z.-W. Sun: [Formula: see text] where p is an odd prime, [Formula: see text] are the Delannoy numbers, and [Formula: see text] are the harmonic numbers. We also prove that, for any positive integer m and prime p > m + 1, [Formula: see text] which is a multiple generalization of Kohnen's congruence. Furthermore, a q-analogue of this congruence is established.

Applications of stuffle product of multiple zeta values

We obtain some formulas for the stuffle product and apply them to derive a decomposition formula for multiple zeta values. Moreover, we give an application to combinatorics and get the following identity:D(n+1,t+1)+D(n,t)=2∑ℓ=0nD(t,n−ℓ)+2∑ℓ=0tD(n,t−ℓ), where D(n,t) is the Delannoy number.

Divisors and specializations of Lucas polynomials

Three-term recurrences have infused stupendous amount of research in a broad spectrum of the sciences, such as orthogonal polynomials (in special functions) and lattice paths (in enumerative combinatorics). Among these are the Lucas polynomials, which have seen a recent true revival. In this paper one of the themes of investigation is the specialization to the Pell and Delannoy numbers. The underpinning motivation comprises primarily of divisibility and symmetry. One of the most remarkable findings is a structural decomposition of the Lucas polynomials into what we term as flat and sharp analogs.

Infinitely log-monotonic combinatorial sequences

We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {an}n⩾0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {an+1/an}n⩾0 is log-concave. Furthermore, we prove that if a sequence {an}n⩾k is ratio log-concave, then the sequence {ann}n⩾k is strictly log-concave subject to an initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers Dn, we confirm a conjecture of Sun on the log-concavity of the sequence {Dnn}n⩾1.

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.

NEW CONGRUENCES FOR SUMS INVOLVING APÉRY NUMBERS OR CENTRAL DELANNOY NUMBERS

The Apéry numbers An and central Delannoy numbers Dn are defined by [Formula: see text] Motivated by some recent work of Z.-W. Sun, we prove the following congruences: [Formula: see text] where n ≥ 1, r ≥ 0, and ε = ± 1. For r = 1, we further show that [Formula: see text] where p > 3 is a prime. The following congruence [Formula: see text] plays an important role in our proof.

Integral polynomial sequences arising from matrix powers of order 2

In this paper we study a recursive system of integers {χ(n,k):n>k⩾0};χ(n+2,k)=χ(n+1,k)-χ(n+1,k-1)+2χ(n,k-1)χ(k+1,k)=-χ(k,k-1)which is uniquely determined by the initial values {χ(n,0)}n=1∞. We show under the constant initial dates χ(n,0)=χ(1,0) for all n that the polynomial χn(x)=∑k=0n-1χ(n,k)xk of degree n-1 is (anti) palindromic. Several explicit formulae for χ(n,k) via Vandermonde matrix, mirrored Γ-matrix, weighed Delannoy number, Riordan array, hypergeometric function, Jacobi polynomial, and some combinatorial identities are derived.