Congruences involving binomial coefficients and Apéry-like numbers

Abstract

For $n=0,1,2,\dots$ let $W_n={\sum }_{k=0}^{[n/3]}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\left(\genfrac{}{}{0ex}{}{3k}{k}\right)\left(\genfrac{}{}{0ex}{}{n}{3k}\right)(-3{)}^{n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then $\{W_n\}$ is an Apéry-like sequence. In this paper we deduce many congruences involving $\{W_n\}$, in particular we determine ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\frac{W_k}{m^k}(\mathrm{mod}p)$ for $m=-640332,-5292,-972,-108,-44,-27,-12,8,54,243$ by using binary quadratic forms, where $p>3$ is a prime. We also prove several congruences for generalized Apéry-like numbers, and pose 29 challenging conjectures on congruences involving binomial coefficients and Apéry-like numbers.

arXiv comment28 pages