# An Aearated Triangular Array of Integers

## TL;DR

This paper introduces a new triangular array of integers related to Bernoulli and Stirling numbers, explores its properties, extensions to negative indices, and proposes a novel generalization of Genocchi numbers, highlighting limited current understanding.

## Contribution

It presents a new integer array linked to Bernoulli and Stirling numbers, extends it to negative indices, and introduces a new generalization of Genocchi numbers with a generating function.

## Key findings

- Array consists of integers and vanishes when n-k is odd.
- Extension to negative indices yields a second kind of integers.
- Proposes a new generalization of Genocchi numbers with a known generating function.

## Abstract

Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli and cycle Stirling numbers. These numbers are all integers and they vanish when $n-k$ is odd. This triangle has many similarities with the Stirling triangle. In particular, we show how it can be extended to negative indices and how this extension produces a {\it second kind} of such integers which may be considered as a new generalization of the Genocchi numbers and for which a generating function is easily obtained. But our knowledge of these integers remains limited, especially for those of the {\it first kind}.