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3.1: Prelude to Generating Functions

  • Page ID
    7152
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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

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

    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 \(\PageIndex{1}\): Generating Function

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

    \[f(x)=\sum_{i=0}^\infty a_i x^i.\nonumber\]

    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.


    This page titled 3.1: Prelude to Generating Functions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.