Generating function ($A(x)={\sum }_{n\ge 0}{a}_n{x}^n$)

A power series encoding a sequence through its coefficients. It turns combinatorial splitting and concatenation rules into algebraic identities. A counting generating function is not necessarily a probability generating function: its coefficients need not sum to one.

An exponential generating function encodes a sequence using the displayed factorial denominators. It is especially useful for counting labelled combinatorial structures: taking an unordered set of structures with positive size corresponds to exponentiating their exponential generating function. The rooted-tree generating function illustrates this through $T=z{e}^T$.