The integrality of the Genocchi numbers obtained through a new identity and other results

TL;DR

This paper explores properties of a specific integer sequence to derive a new identity for Genocchi numbers, proving their integrality, and also establishes lower bounds for 2-adic valuations of certain sums.

Contribution

It introduces a new identity linking a sequence to Genocchi numbers, demonstrating their integrality, and provides bounds on 2-adic valuations of related sums.

Findings

Genocchi numbers are integers for all n.

A new identity connecting a sequence to Genocchi numbers is established.

Lower bounds for 2-adic valuations of sums involving powers of 2 are derived.

Abstract

In this note, we investigate some properties of the integer sequence of general term $a_n := \sum_{k = 0}^{n - 1} k! (n - k - 1)!$ ($\forall n \geq 1$) to derive a new identity of the Genocchi numbers $G_n$ ($n \in \mathbb{N}$), which immediately shows that $G_n \in \mathbb{Z}$ for any $n \in \mathbb{N}$. In another direction, we obtain nontrivial lower bounds for the $2$-adic valuations of the rational numbers $\sum_{k = 1}^{n} \frac{2^k}{k}$.