Congruences for Catalan-Larcombe-French numbers

Abstract

Let $\{P_n\}$ be the Catalan-Larcombe-French numbers given by $P_0=1,P_1=8$ and $n^2{P}_n=8(3n^2-3n+1)P_{n-1}-128(n-1{)}^2{P}_{n-2}$ $(n\ge 2)$, and let $S_n=P_n/2^n$. In this paper we deduce congruences for $S_{m{p}^r}(\mathrm{mod}{p}^{r+2})$, $S_{m{p}^r-1}(\mathrm{mod}{p}^r)$ and $S_{m{p}^r+1}(\mathrm{mod}{p}^{2r})$, where $p$ is an odd prime and $m,r$ are positive integers. We also prove that $S_{(p^2-1)/2}\equiv 0(\mathrm{mod}{p}^2)$ for any prime $p\equiv 5,7(\mathrm{mod}8)$, and show that $\{S_m\}$ is log-convex.

26 pages

References in corpus (3)

• Lucas' theorem: its generalizations, extensions and applications (1878--2014)
• Super congruences involving Bernoulli and Euler polynomials
• Congruences involving Franel and Catalan-Larcombe-French numbers