Consider a matrix $A$ of size $n\times n$. I want to fill it with one and zero such that there are exactly two entries one in each row and each column, and the other entries are zero.
In how many different ways I can fill $A$?
You find your series in the online encyclopedia of integer series.
Note that the given formulas are all recursive, sums or asymptotic results.
In particular, the generating function is related to the generating functions of derangements, permutations without fixed points.