So far, I have found (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind:
$$ \sum_{l=1}^{n} |S_{1}(n,l)|z^{l} = (z)_{n} = \prod_{k=0}^{n-1} (z+k) , $$ and $$ \sum_{n=l}^{\infty} \frac{|S_{1}(n,l)|}{n!}z^{n} = (-1)^{l} \frac{\ln^{l}(1-z)}{l!} .$$
Instead of summing the latter expression over $n$, I'm curious whether there is a closed form of the generating sum when summed over the other indices:
$$ f(z) := \sum_{k=1}^{n} \frac{|S_{1}(n,k)|}{k!}z^{k} .$$ One could also take the sum from $k=1$ to $k=\infty$, as $|S_{1}(n,k)| = 0$ when $k>n$.
Question: is there a closed form or succinct formula for $f(\cdot)$ ?
Both derivations and references would be much appreciated.