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Congruences for Apéry numbers $β_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$ (1812.10351v5)

Published 26 Dec 2018 in math.NT

Abstract: In this paper we establish some congruences involving the Ap\'ery numbers $\beta_{n}=\sum_{k=0}{n}\binom{n}{k}2\binom{n+k}{k}$ $(n=0,1,2,\ldots)$. For example, we show that $$\sum_{k=0}{n-1}(11k2+13k+4)\beta_k\equiv0\pmod{2n2}$$ for any positive integer $n$, and $$\sum_{k=0}{p-1}(11k2+13k+4)\beta_k\equiv 4p2+4p7B_{p-5}\pmod{p8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of A\'pery numbers, $\beta_n$ and $A_n=\sum_{k=0}n\binom nk2\binom{n+k}k2$.

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