We present several series involving central binomial coefficients, Catalan numbers \(C_n\) , harmonic numbers \(H_n\) , odd harmonic numbers \(O_n\) , and various products of these classical sequences. A central highlight of our analysis is the derivation of closed-form expressions for two notable classes of series: \(\sum \nolimits _{n=1}^\infty \left( {\begin{array}{c}2n\\ n\end{array}}\right) H_nO_n x^n\) and \(\sum \nolimits _{n=1}^\infty C_nH_nO_nx^n\) . Our results build upon and complement the work of Lehmer, Boyadzhiev, Chen, Li and Chu, Sofo, and several other researchers in this area of analytic and combinatorial number theory. In addition, we introduce new families of series involving products of harmonic and related numbers, some of which yield elegant and unexpected identities. Some of our series cannot be expressed using elementary functions and instead involve combinations of dilogarithm functions. Only in exceptional cases—such as when the golden ratio appears—can they be written in simple closed forms. These findings shed new light on the rich connections between combinatorics and classical analysis.
