I like generating functions. They provide quick solutions to some nasty recurrence relations. Sometimes even when there is no closed form for the sequence, one can find pretty functional equations for the associated g.f.
My question is rather general. Suppose there is a sequence $a_n$ for natural $n$ describing some set cardinality, as it is usual in combinatorics. Suppose also that we know a closed or integral form $a(x)$ of the corresponding generating function. Is there any relevant information about the underlying problem we can retrieve just looking at the generating function? Other than the terms of the sequence and asymptotic behavior, of course.