Congruences for Apéry numbers ${\beta }_n={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$

Hui-Qin Cao, Yuri Matiyasevich, Zhi-Wei Sun

In this paper we establish some congruences involving the Apéry numbers ${\beta }_n={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$ $(n=0,1,2,\dots )$. For example, we show that $$\sum_{k=0…