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.


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