An alternative recursive formula for the sums of powers of integers

TL;DR

This paper introduces a new recursive formula for sums of powers of integers using Stirling numbers and offers a novel non-recursive definition of Catalan numbers, contributing to combinatorial mathematics.

Contribution

It presents an alternative recursive formula involving Stirling numbers and a new non-recursive definition of Catalan numbers, enhancing mathematical tools for combinatorics.

Findings

Derived a recursive formula for sums of powers using Stirling numbers

Provided a non-recursive definition of Catalan numbers

Offers new insights into combinatorial identities

Abstract

In this note, we derive an alternative recursive formula for the sums of powers of integers involving the Stirling numbers of the first kind. As a remarkable by-product, we provide a non-recursive definition of the Catalan numbers.