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Congruences involving binomial coefficients and Apéry-like numbers (1803.10051v7)
Published 27 Mar 2018 in math.NT and math.CO
Abstract: For $n=0,1,2,\ldots$ let $W_n=\sum_{k=0}{[n/3]}\binom{2k}k \binom{3k}k\binom n{3k}(-3){n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then ${W_n}$ is an Ap\'ery-like sequence. In this paper we deduce many congruences involving ${W_n}$, in particular we determine $\sum_{k=0}{p-1}\binom{2k}k\frac{W_k}{mk}\pmod p$ 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 Ap\'ery-like numbers, and pose 29 challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.