It is shown here on Mathworld's page on Stirling number of the second kind that

$$ \sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z}) $$

where $S(n,k)$ is Stirling number of the second kind and $\text{Li}_{1-n}$ is the polylogarithm.

Can somebody provide me some reference on where this identity came from? It isn't shown on Mathworld's page.