On some divisibility properties of binomial sums

TL;DR

This paper investigates specific binomial sums inspired by series for 1/π², proving they are divisible by a quadratic multiple of central binomial coefficients, thus extending known divisibility results.

Contribution

It establishes stronger divisibility properties for two binomial sums related to series for 1/π², surpassing previous results by He using the WZ-method.

Findings

Proves sums are divisible by 2n^2 * binom(2n, n)^2 for all n ≥ 2.

Extends previous divisibility results by demonstrating stronger divisibility properties.

Provides new insights into the divisibility structure of binomial sums related to π² series.

Abstract

In this paper, we consider two particular binomial sums \begin{align*} \sum_{k=0}^{n-1}(20k^2+8k+1){\binom{2k}{k}}^5 (-4096)^{n-k-1} \end{align*} and \begin{align*} \sum_{k=0}^{n-1}(120k^2+34k+3){\binom{2k}{k}}^4\binom{4k}{2k} 65536^{n-k-1}, \end{align*} which are inspired by two series for $\frac{1}{\pi^2}$ obtained by Guillera. We consider their divisibility properties and prove that they are divisible by $2n^2 \binom{2n}{n}^2$ for all integer $n\geq 2$. These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by $2n \binom{2n}{n}$ with the WZ-method.