Combinatorial remarks about a "remarkable identity"

In the June 2012 issue of this Magazine, Frumosu and Teodorescu-Frumosu proved that, for all integers m ≥ 2, where the inner sum is taken over all p-term ordered partitions of m. Their proof is calculus-based, relying on power series manipulations. In this note, we provide a combinatorial proof of this identity (which they requested at the end of their article) and we use the insights gained via this argument to prove other results of a similar type.