Exponential generating function ($A(z)={\sum }_{n\ge 0}{a}_n{z}^n/n!$)

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 $T=z{e}^T$.