If $g(z)$ is analytic near $0$, and $g(z) = \exp(z\,g(z))$, then $g(z)=\sum_{n \geq 0}\frac{(n+1)^{n-1}}{n!} z^n$ near $0$

Question

This is a question from the article "An Occupancy Discipline and Applications".

$g(z)$ is an analytic function near $0$. Suppose $$g(z) = \exp(z\,g(z))$$ Then prove that, near $z=0$, $$g(z) = \sum_{n \geq 0} \frac{(n+1)^{n-1}}{n!} z^n$$

I have tried to read the reference "Solution of the equation ze^z = a", but I can not find a solution for this equation.