Corpus ID: 119141960

Congruences for Apéry numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$

@article{Cao2018CongruencesFA,
  title={Congruences for Ap{\'e}ry numbers \$\beta\_\{n\}=\sum\_\{k=0\}^\{n\}\binom\{n\}\{k\}^2\binom\{n+k\}\{k\}\$},
  author={H. Cao and Y. Matiyasevich and Z. Sun},
  journal={arXiv: Number Theory},
  year={2018}
}
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 $$\sum_{k=0}^{n-1}(11k^2+13k+4)\beta_k\equiv0\pmod{2n^2}$$ for any positive integer $n$, and $$\sum_{k=0}^{p-1}(11k^2+13k+4)\beta_k\equiv 4p^2+4p^7B_{p-5}\pmod{p^8}$$ 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… Expand
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Figures from this paper

Two congruences concerning Apéry numbers

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