A generating function that produces $\{n^n\}_{n=0}^{\infty}$

Question

I am trying to find a generating function that produces the sequence $$ 1, 2^2, 3^3, 4^4, 5^5,\cdots,n^n $$ so that $$ F(x) = \sum_{n=0}^{\infty}n^n\frac{x^n}{n!} $$ Does anyone know of such a generating function? I have been reading the book generatingfunctionology but I cannot find such a function or how to derive such a function.

Answer 1

Here is a generating function

$$ F(x)=1+xW'_0(-x) .$$

where $W(x)$ is the Lambert $W$-function. Here are few terms of the series of $F(x)$

$$ 1+xW'_0(-x) = 1+x+2\,{x}^{2}+{\frac {9}{2}}{x}^{3}+{\frac {32}{3}}{x}^{4}+{\frac { 625}{24}}{x}^{5}+O( {x}^{6}), $$

Note: For the proof, just use the power series representation

$$ W_0 (x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}\ x^n . $$