A summation with Stirling numbers of the Second kind Can the following sum be evaluated?
$$\sum\limits_{k=1}^{n-1}\binom{n}{n-k}\left( n-k\right)
!S\left( n,n-k\right) \left( \frac{k-1}{k}\right) ^{k}$$
where $S(n,m)$ is the Stirling number of the second kind.

Hint:
use :
$$n^n=\sum_{k=1}^{n}\binom{n}{n-k}\left(n-k\right)!{n\brace n-k}$$
And observe that ${n\brace 0}=0$ for $n>0$
