Congruences involving Delannoy numbers and Schr\"oder numbers

Abstract

The central Delannoy numbers $D_n=\sum_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}$ and the little Schr\"oder number $s_n=\sum_{k=1}^{n}\frac{1}{n}\binom{n}{k}\binom{n}{k-1}2^{n-k}$ are important quantities. In this paper, we confirm \[\frac{2}{3n(n+1)}\sum_{k=1}^n (-1)^{n-k}k^2D_kD_{k-1}\ \text{and}\ \ \frac 1n\sum_{k=1}^n (-1)^{n-k}(4k^2+2k-1)D_{k-1}s_k\]are positive odd integers for all $n=1,2,3,\cdots$. We also show that for any prime number $p>3$, \[\sum_{k=1}^{p-1} (-1)^kk^2D_kD_{k-1}\ \equiv\ -\frac56p \pmod{p^2}\] and \[\sum_{k=1}^p (-1)^k(4k^2+2k-1)D_{k-1}s_k\ \equiv\ -4p \pmod{p^2}\text{.}\] Moreover, define \begin{equation*} s_n(x)=\sum_{k=1}^{n}\frac{1}{n}\binom{n}{k}\binom{n}{k-1}x^{k-1}(x+1)^{n-k}, \end{equation*} for any $n\in\mathbb{Z}^+$ is even we have \begin{equation*} \frac{4}{n(n+1)(n+2)(1+2x)^3}\sum_{k=1}^{n}k(k+1)(k+2)s_k(x)s_{k+1}(x)\in\mathbb{Z}[x]. \end{equation*}