Three combinatorial sums involving central binomial coefficients
Kunle Adegoke, Robert Frontczak
TL;DR
This work analyzes three combinatorial sums built from central binomial coefficients and harmonic numbers, namely $S_r^H(n)$, $S_r^O(n)$, and $S_r^{2H}(n)$. It develops recursive frameworks for these sums via summation by parts and connects them to the base family $S_r(n)$, while also providing an alternative closed-form route using Stirling numbers and $r$-Stirling numbers of the second kind. The authors derive explicit formulas for $S_0^O(n)$ and $S_0^H(n)$ and present recursive relations for $S_r^O(n)$, $S_r^H(n)$, and $S_r^{2H}(n)$, including closed forms for small $r$ expressed in terms of $O_{n+1}$ and $H_{n+1}$. An additional approach expresses $S_r^O(n)$ in closed form via Stirling and $r$-Stirling numbers, with concrete special cases such as $S_0^O(n)$ and $S_1^O(n)$, using Gould-type polynomial identities. Together, the results furnish recursive tools and explicit closed forms for central-binomial-sum identities with harmonic components, enhancing analytic and combinatorial understanding of these sums.
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.