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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 (nk), P(n,k), or the number of derangements of [n]. Also recall that

(x+1)n=nk=0(nk)xk.

This provides the values (nk) as coefficients of the Maclaurin expansion of a function. This turns out to be a useful idea.

Definition 3.1.1: Generating Function

f(x) is a generating function for the sequence a0,a1,a2, if

f(x)=i=0aixi.

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 4.0 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.

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