As said within the comment section it is sufficient to visit the Wikipedia Page and therefore no need to invoke some kind of Harmonic Numbers here. Within the subsection Generating Functions we eventually find the following paragraph:
A variety of identities may be derived by maniplulating the generating function:
\begin{align*}
H(z,u)=(1+z)^u&=\sum_{n=0}^\infty\binom unz^n\\
&=\sum_{n=0}^\infty\frac{z^n}{n!}\sum_{k=0}^ns(n,k)u^k\\
&=\sum_{k=0}^\infty u^k\sum_{n=k}^\infty\frac{z^n}{n!}s(n,k)
\end{align*}
Using the equality
$$(1+z)^u=e^{u\log(1+z)}=\sum_{k=0}^\infty(\log(1+z))^k\frac{u^k}{k!}$$
it follows that
$$\sum_{n=k}^\infty(-1)^{n-k}\begin{bmatrix}n\\k\end{bmatrix}\frac{z^n}{n!}=\frac{(\log(1+z))^k}{k!}$$
The crucial relations used here are
\begin{align*}
&1.&&(x)_n~=~\sum_{k=0}^n s(n,k)x^k\\
&2.&&s(n,k)~=~(-1)^{n-k}\begin{bmatrix}n\\k\end{bmatrix}
\end{align*}
Which are, as far as I can tell (not being that experienced with Stirling Numbers at all), quite fundamental properties of the Stirling Numbers of the First Kind.
