Given a generating function for some sequence, I'm basically interested in the first few values. Well an explicit closed form would be nice, but often there isn't any. I suppose if there is any, I'd try to find the Taylor expansion at $0$ to get the coefficients, right? So I have to be able to calculate the n-th derivative?
But also if there is no explicit closed form a recursive formula would be nice. For a computer it shouldn't be too difficult to calculate some of the first few values then. Is it possible to derive such a recursion directly from the generating function?
The formal method provides good cooking recipes for translating a combinatorial class into a generating function (I'm referring here to Flajolet's and Sedgewick's terminology from Analytic Combinatorics). Is there something similar which yields a recursion instead of a generating function?
I couldn't find anything about recursions, except how to solve recursions by use of generating functions. But actually I find the opposite useful too.