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Mathematics LibreTexts

3.1: Prelude to Generating Functions

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    As we have seen, a typical counting problem includes one or more parameters, which of course show up in the solutions, such as \(n\choose k\), \(P(n,k)\), or the number of derangements of \([n]\). Also recall that

    $$(x+1)^n=\sum_{k=0}^n {n\choose k}x^k.$$

    This provides the values \({n\choose k}\) as coefficients of the Maclaurin expansion of a function. This turns out to be a useful idea.

    Definition: generating function

    \(f(x)\) is a generating function for the sequence \(a_0,a_1,a_2,\ldots\) if

    $$f(x)=\sum_{i=0}^\infty a_i x^i.$$

    Sometimes a generating function can be used to find a formula for its coefficients, but if not, it gives a way to generate them. Generating functions can also be useful in proving facts about the coefficients.