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arXiv · 1502.02499

Congruences involving Franel and Catalan-Larcombe-French numbers

Zhi-Hong Sun
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Abstract

Let $\{f_n\}$ be the Franel numbers given by $f_n=\sum_{k=0}^n\binom nk^3$, and let $p>5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$ for $m=5,-16,16,32,-49,50,96$. Let $S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$. We also determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$ for $m=7,16,25,32,64,160,800,1600, 156832$.

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Zhi-Hong Sun. 2015-05-04. Congruences involving Franel and Catalan-Larcombe-French numbers. https://arxiv.org/abs/1502.02499

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