A supercongruence involving Delannoy numbers and Schr\"oder numbers

Abstract

The Delannoy numbers and Schr\"oder numbers are given by \begin{align*} D_n=\sum_{k=0}^n{n\choose k}{n+k\choose k}\quad \text{and}\quad S_n=\sum_{k=0}^n{n\choose k}{n+k\choose k}\frac{1}{k+1}, \end{align*} respectively. Let $p>3$ be a prime. We mainly prove that \begin{align*} \sum_{k=1}^{p-1}D_k S_k\equiv 2p^3B_{p-3}-2pH^{*}_{p-1} \pmod{p^4}, \end{align*} where $B_n$ is the $n$-th Bernoulli number and those $H^{*}_n$ are the alternating harmonic numbers given by $H^{*}_n=\sum_{k=1}^{n}\frac{(-1)^k}{k}$. This supercongruence was originally conjectured by Z.-W. Sun in 2011.