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
\[(x+1)^n=\sum_{k=0}^n {n\choose k}x^k.\nonumber \]
This provides the values \({n\choose k}\) as coefficients of the Maclaurin expansion of a function. This turns out to be a useful idea.
\(f(x)\) 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.