# Deriving recursion from generating function

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.

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If you know the generating function, you can convert it into a recursion if you can recognize it as the solution to some algebraic/differential equation in a sort of backwards way from how one goes from a recursive formula to a generating function. Unfortunately, I don't know any general way to find relations satisfied by complicated formulae. –  Aaron Dec 26 '11 at 19:25
Well, for example if you are given the function and if you know that your generating function satisfies some finite order differential equation, you may try using linear algebra to find a differential equation that it satisfies. –  Aleks Vlasev Dec 26 '11 at 19:35