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For example in my textbook it says that the sequence for the generating function

$$f(x) = (2x-3)^3$$

is

$$-27,54,-36,8,0,0,0,\dots$$

I can see that plugging $0$ into the function gives me $-27$, but this doesn't hold true for $1$ and $54$. How do I get this sequence or the correct sequence for another generating function like

$$f(x) = \frac{x^3}{1-x^2}\;?$$

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    To compute the term $a_{n}$ in the sequence generated by an ordinary generating function, take the $n$-th derivative of the function at $x=0$ and divide by $n!$.2012-12-05

4 Answers 4