Three combinatorial sums involving central binomial coefficients

TL;DR

This paper investigates three types of combinatorial sums involving central binomial coefficients and harmonic numbers, deriving recursive formulas and closed-form expressions using summation by parts and Stirling numbers.

Contribution

It introduces new recursive formulas and closed-form expressions for sums involving binomial coefficients and harmonic numbers, using summation by parts and Stirling numbers.

Findings

Derived recursive expressions for the sums.

Provided closed-form expressions in terms of Stirling numbers.

Offered an alternative approach for sum representation.

Abstract

We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive expressions for these sums. In addition, we offer an alternative approach to express one class of sums and some related sums in closed form in terms of Stirling numbers and r-Stirling numbers of the second kind.