\begin{definition}[Definition:Exponential Generating Function]
Let $A = \sequence {a_n}$ be a sequence in $\R$.
Then $\ds \map {G_A} z = \sum_{n \mathop \ge 0} \frac {a_n} {n!} z^n$ is called the '''(exponential) generating function''' for the sequence $A$.
The mapping $\map {G_A} z$ is defined for all $z$ for which the power series $\ds \sum_{n \mathop \ge 0} \frac {a_n} {n!} z^n$ is convergent.
The definition can be modified so that the lower limit of the summation is $b$ where $b > 0$ by assigning $a_k = 0$ where $0 \le k < b$.
\end{definition}