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Given a generating function $F(z)$, is there any way to compute the coefficient $a_k$ of $z^k$ by Mathematica? One method is to calculate by the equation $\frac{F^{(k)}(0)}{k!}$. However, this approach is not feasible for $k$ large.

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Look into SeriesCoefficient. –  RiemannZeta Oct 17 '13 at 3:16
    
@RiemannZeta do you have any idea of how they implemented it? I mean any bound for the deviation/approximation? –  Loi.Luu Oct 17 '13 at 3:30
    
Section 7.3 of Concrete Mathematics by Graham, Knuth, and Patashnik (May 1989 edition) describes in detail what I suspect is buried in SeriesCoefficient. –  KennyColnago Nov 4 '13 at 6:25
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