Congruences involving Delannoy numbers and Schr\"oder numbers

TL;DR

This paper investigates congruences and positivity properties of sums involving Delannoy and Schr"oder numbers, establishing new divisibility, congruence, and polynomial integrality results.

Contribution

It provides new positivity results, congruences modulo primes, and polynomial integrality properties for sums involving Delannoy and Schr"oder numbers.

Findings

Certain sums are positive odd integers for all n

Specific sums satisfy congruences modulo p^2 for primes p>3

A polynomial expression involving s_k(x) is an integer polynomial in x

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*}