Factors of sums and alternating sums involving binomial coefficients and powers of integers

TL;DR

This paper investigates divisibility properties of sums involving binomial coefficients and powers of integers, proving specific congruences and proposing conjectures for related sums with alternating signs.

Contribution

It establishes new divisibility results for sums of binomial coefficients with powers and introduces conjectures extending these properties.

Findings

Proved divisibility of certain sums involving binomial coefficients and odd powers.

Established congruences for sums with alternating signs and powers.

Proposed conjectures for broader classes of sums with similar structures.

Abstract

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$ \sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}}, $$ where $\epsilon=\pm 1$.