On 2-adic orders of some binomial sums

TL;DR

This paper proves a divisibility property of certain binomial sums involving powers of integers, confirming a recent conjecture and revealing the 2-adic order related to binary representations.

Contribution

It establishes a new divisibility result for binomial sums involving powers, confirming a conjecture by Guo and Zeng.

Findings

The binomial sum is divisible by 2^{2n - min(α(n), α(r))}.

The result links 2-adic valuation to binary digit counts.

Confirms a conjecture on 2-adic orders of binomial sums.

Abstract

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$ \sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.