Exponential generating function

Definition

Let $K$ be a field of characteristic $0$, for example $\mathbb{Q}$ or $\mathbb{R}$, and let $(a_n)_{n \ge 0}$ be a sequence in $K$. The exponential generating function of $(a_n)$ is the formal power series $$\hat{A}(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!} \in K[\![x]\!].$$

Remarks

The constant sequence $1$ has the exponential generating function $e^x = \sum_n x^n / n!$. For exponential generating functions, the product corresponds to the binomial convolution $c_n = \sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$. They suit labelled structures. For example, the derangement numbers have the exponential generating function $\frac{e^{-x}}{1 - x}$.

Explanations

L1 human

An exponential generating function divides the $n$-th term by $n!$ before it puts it in the series. This makes the series of fast-growing sequences, such as $n!$, easy to handle: the sequence $n!$ gives $\frac{1}{1 - x}$.