Congruences involving $\binom{4k}{2k}$ and $\binom{3k}k$

TL;DR

This paper investigates specific binomial sum congruences modulo a prime greater than 3, extending understanding of combinatorial identities in number theory.

Contribution

It determines new congruences for sums involving binomial coefficients with parameters 3k and 4k modulo primes greater than 3.

Findings

Explicit formulas for sums involving inom{4k}{2k} and inom{3k}{k} modulo p

Results for sums with alternating signs and powers of -3

Extensions of known binomial sum congruences

Abstract

Let $p$ be a prime greater than 3. In the paper we mainly determine $\sum_{k=0}^{[p/4]}\binom{4k}{2k}(-1)^k$, $\sum_{k=0}^{[p/3]}\binom{3k}k, \sum_{k=0}^{[p/3]}\binom{3k}k(-1)^k$ and $\sum_{k=0}^{[p/3]}\binom{3k}k(-3)^k$ modulo $p$, where $[x]$ is the greatest integer not exceeding $x$.