Im looking for functions $g(z)$ such that
$$g(z) = \sum_{n=0}^{\infty} \frac{g(n)}{(n+1)!} z^n = g(0) + \frac{g(1)}{2} z + \frac{g(2)}{6} z^2 + \frac{g(3)}{24} z^3 + ...$$
and $g(n)$ are all positive reals; $0 \leq g(n)$.
And I wonder how fast $g(n)$ grows.
I got inspired by this :
$f(z) = \sum_{n=0}^{\infty} f(n)^2 z^n$?
But since that did not behave very well - see the edit and the comments there - , I made this variant.
I am not sure this works out nice, Im just wondering.
For clarity with "nice " or behaving "good" I mean being an entire function.