On a curious integer sequence

TL;DR

This paper investigates a specific recursive sequence, proving it consists of integers by linking it to Genocchi and Stirling numbers, and explores its properties and related open problems.

Contribution

It establishes the integrality of the sequence by expressing it through Genocchi and Stirling numbers, providing new insights into its structure.

Findings

The sequence is composed of integers.

Explicit expression of the sequence in terms of known number sequences.

Several corollaries and open problems related to the sequence.

Abstract

This note is devoted to study the recurrent numerical sequence defined by: $a_0 = 0$, $a_n = \frac{n}{2} a_{n - 1} + (n - 1)!$ ($\forall n \geq 1$). Although, it is immediate that ${(a_n)}_n$ is constituted of rational numbers with denominators powers of $2$, it is not trivial that ${(a_n)}_n$ is actually an integer sequence. In this note, we prove this fact by expressing $a_n$ in terms of the Genocchi numbers and the Stirling numbers of the first kind. We derive from our main result several corollaries and we conclude with some remarks and open problems.