HMMT 2020 U23

A function $f: A \rightarrow A$ is called idempotent if $f(f(x))=f(x)$ for all $x \in A$. Let $I_{n}$ be the number of idempotent functions from $\{1,2, \ldots, n\}$ to itself. Compute

$$\sum_{n=1}^{\infty} \frac{I_{n}}{n!}$$