Two Triple binomial sum supercongruences

TL;DR

This paper proves a new supercongruence involving triple binomial sums, extending previous conjectures and introducing a novel enumeration of abelian squares, with implications for combinatorial number theory.

Contribution

It provides a proof for a specific supercongruence conjecture and introduces a new combinatorial enumeration related to abelian squares.

Findings

Proved a new supercongruence involving triple binomial sums.

Established a connection between supercongruences and combinatorial enumeration.

Proposed further conjectures and remarks on related supercongruences.

Abstract

In a recent article, Apagodu and Zeilberger (http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating abelian squares, and we leave some remarks for the others.