while studying a problem about double factorial computation I faced the following (conjectured) identity
$$\sum_{k=1}^{n-1}(-1)^{n-1-k}{n-1 \choose k}k^n = \frac{n-1}{2}n! $$
I just tried small values of $n>1$ and noticed that the right side is a Lah number. I have and found no proof: any help ?
Thank you very much in advance.