# Logarithms of a binomial series: A Stirling number approach

## TL;DR

This paper introduces a novel approach to compute powers of the logarithm of the Catalan generating function using Stirling cycle numbers, offering an alternative expression in terms of harmonic numbers.

## Contribution

It presents a new method for calculating logarithmic powers of the Catalan generating function via Stirling cycle numbers and harmonic numbers.

## Key findings

- Derived explicit formulas using Stirling cycle numbers.
- Expressed the generating function in terms of higher order harmonic numbers.
- Provided a new combinatorial approach to analyze Catalan-related functions.

## Abstract

The $p$-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers. Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.