New congruences for sums involving Apery numbers or central Delannoy numbers

TL;DR

This paper establishes new congruences involving sums of Apéry and central Delannoy numbers, extending previous work and revealing deeper modular properties of these combinatorial sequences.

Contribution

The paper proves novel congruences for sums involving Apéry and Delannoy numbers, including higher power modulus results and connections to recent conjectures.

Findings

Sum of (2k+1)^{2r+1} times Apéry or Delannoy numbers is divisible by n.

Special case for r=1 shows sum divisible by n^3 and a prime-specific congruence.

Key role of a specific binomial sum congruence in the proofs.

Abstract

The Ap\'ery numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\sum_{k=0}^{n}{n+k\choose 2k}^2{2k\choose k}^2, \quad D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences: \sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\equiv \sum_{k=0}^{n-1}\varepsilon^k (2k+1)^{2r+1}D_k \equiv 0\pmod n, where $n\geqslant 1$, $r\geqslant 0$, and $\varepsilon=\pm1$. For $r=1$, we further show that \sum_{k=0}^{n-1}(2k+1)^{3}A_k &\equiv 0\pmod{n^3}, \quad \sum_{k=0}^{p-1}(2k+1)^{3}A_k &\equiv p^3 \pmod{2p^6}, where $p>3$ is a prime. The following congruence \sum_{k=0}^{n-1} {n+k\choose k}^2{n-1\choose k}^2 \equiv 0 \pmod{n} plays an important role in our proof.