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Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.
A generating function is a formal power series of the form \begin{equation*} f(x)=\sum^{\infty}_{n=0}a_nx^n \end{equation*} whose coefficients contain information about $a_n$, the sequence of numbers.