3.1: Prelude to Generating Functions

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.