Congruences involving Franel and Catalan-Larcombe-French numbers

Abstract

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

25 pages

References in corpus (1)

• Super congruences involving Bernoulli and Euler polynomials

Cited by in corpus (1)

• Congruences for Catalan-Larcombe-French numbers