Arithmetic properties of Delannoy numbers and Schr\"oder numbers

Abstract

Define $$D_n(x)=\sum_{k=0}^n\binom nk^2x^k(x+1)^{n-k}\ \ \ \mbox{for}\ n=0,1,2,\ldots$$ and $$s_n(x)=\sum_{k=1}^n\frac1n\binom nk\binom n{k-1}x^{k-1}(x+1)^{n-k}\ \ \ \mbox{for}\ n=1,2,3,\ldots.$$ Then $D_n(1)$ is the $n$-th central Delannoy number $D_n$, and $s_n(1)$ is the $n$-th little Schr\"oder number $s_n$. In this paper we obtain some surprising arithmetic properties of $D_n(x)$ and $s_n(x)$. We show that $$\frac1n\sum_{k=0}^{n-1}D_k(x)s_{k+1}(x)\in\mathbb Z[x(x+1)]\ \quad\mbox{for all}\ n=1,2,3,\ldots.$$ Moreover, for any odd prime $p$ and $p$-adic integer $x\not\equiv0,-1\pmod p$, we establish the supercongruence $$\sum_{k=0}^{p-1}D_k(x)s_{k+1}(x)\equiv0\pmod{p^2}.$$ As an application we confirm Conjecture 5.5 in [S14a], in particular we prove that $$\frac1n\sum_{k=0}^{n-1}T_kM_k(-3)^{n-1-k}\in\mathbb Z\quad\mbox{for all}\ n=1,2,3,\ldots,$$ where $T_k$ is the $k$-th central trinomial coefficient and $M_k$ is the $k$-th Motzkin number.