Related papers: Congruences concerning generalized central trinomi…

For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and…

A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial…

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central…

Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3…

For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the…

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is…

Let $M_n$ and $T_n$ denote the $n$th Motzkin number and the $n$th central trinomial coefficient respectively. We prove that for any prime $p\ge 5$, \begin{align*} &\sum_{k=0}^{p-1}M_k^2\equiv…

It is shown how to obtain an asymptotic expansion of the generalised central trinomial coefficient $[x^n](x^2 + bx + c)^n$ by means of singularity analysis, thus proving a conjecture of Zhi-Wei Sun.

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related…

The divisibility and congruence of usual and generalized central trinomial coefficients have been extensively investigated. The present paper is devoted to analytic properties of these numbers. We show that usual central trinomial…

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for…

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for…

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations.…

Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and $\chi_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of…

We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a, b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a, b}_n\bmod p$ (with $n<p$). We apply our…

Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of…

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…

Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p.…

The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study…