It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product
More work has to be done if one wants to get asymptotic estimates of the function $p(n)$ (see for instance the pioneering work of Hardy and Ramanujan).
My question is the following: in the course of my research I have found the following generating function (which has a very similar aspect):
My question is if there are kwnown "natural" combinatorial families enumerated by this generating function.
If yes, I will be happy to know some references.