# How can I tweak a generating function so as to multiply its coefficients by the factorial of the term number?

I have the following generating function, $$e^{x e^x}$$, whose first 5 coefficients are the following: $$\{ 1, 1, \frac{3}{2}, \frac{5}{3}, \frac{41}{24} \}$$. These coefficients are almost what I want except for the fact that they should be multiplied by the factorial of the term number i.e. the $$n^{\text{th}}$$ coefficient should be multiplied by $$n!$$, producing the following coefficients: $$\{ 1, 1, 3, 10, 41 \}$$.

I understand that to get these coefficients, we may call $$e^{x e^x}$$ an exponential generating function, and then it is implicitly understood that we must multiply each coefficient by $$n!$$, but is there a way to rewrite the generating function to produce these coefficients directly?

For context, the $$n^{\text{th}}$$ coefficient of this generating function is the the number of idempotent endofunctions for a size of size n.

I'm not too familiar with generating functions, so please correct me if I'm interpreting something incorrectly.