Congruences involving binomial coefficients and Ap\'ery-like numbers

TL;DR

This paper investigates congruences involving Apéry-like sequences, especially the sequence W_n, and establishes new results and conjectures using advanced number theory techniques such as binary quadratic forms.

Contribution

It derives numerous new congruences for Apéry-like numbers, including explicit sums modulo primes for specific parameters, and introduces 29 conjectures on related binomial coefficient congruences.

Findings

Determined sums involving W_n modulo primes for specific m values.

Proved several congruences for generalized Apéry-like numbers.

Posed 29 new conjectures on binomial coefficient congruences.

Abstract

For $n=0,1,2,\ldots$ let $W_n=\sum_{k=0}^{[n/3]}\binom{2k}k \binom{3k}k\binom n{3k}(-3)^{n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then $\{W_n\}$ is an Ap\'ery-like sequence. In this paper we deduce many congruences involving $\{W_n\}$, in particular we determine $\sum_{k=0}^{p-1}\binom{2k}k\frac{W_k}{m^k}\pmod p$ for $m=-640332,-5292,-972,-108,-44,-27,-12,8,54,243$ by using binary quadratic forms, where $p>3$ is a prime. We also prove several congruences for generalized Ap\'ery-like numbers, and pose 29 challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.