For integers b and c the generalized central trinomial coefficient T n (b, c) denotes the
coefficient of xn in the expansion of (x 2+ bx+ c) n. Those T n= T n (1, 1)(n= 0, 1, 2,…) are the …

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of
terms involving products of three binomial coefficients. Let $ p> 3$ be a prime. We prove that …

The Apéry polynomials are given by (Those An= An (1) are Apéry numbers.) Let p be an odd
prime. We show that and that for any p-adic integer x≢ 0 (modp). This enables us to …

The nth Delannoy number and the nth Schröder number given by respectively arise
naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that and …

A generalized central trinomial coefficient T n (b, c) is the coefficient of xn in the expansion of
(x 2+ bx+ c) n (x^ 2+ bx+ c)^ n with b, c∈ ℤ b, c ∈ Z. In this paper we investigate …

The Ap\'ery numbers $ A_n $ and central Delannoy numbers $ D_n $ are defined by $$
A_n=\sum_ {k= 0}^{n}{n+ k\choose 2k}^ 2 {2k\choose k}^ 2,\quad D_n=\sum_ {k= 0}^{n}{n+ …

In the paper, by a general and fundamental, but non-extensively circulated, formula for
derivatives of a ratio of two differentiable functions and by a recursive relation of the …

Full article: Some congruences involving powers of Legendre polynomials Skip to Main
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A new recursive formula arising from a determinantal expression for weighted Delannoy
numbers Page 1 Turkish Journal of Mathematics Volume 45 Number 1 Article 28 1-1-2021 …

Abstract The Delannoy numbers and Schröder numbers are given by D n=∑ k= 0 nnkn+ kk
and S n=∑ k= 0 nnkn+ kk 1 k+ 1, respectively. We mainly prove that for any prime p≥ 5,∑ …