Generating function for $\sum_{k\geq 1} H^{(k)}_n x^ k $ Is there a generating function for 
$$\tag{1}\sum_{k\geq 1} H^{(k)}_n x^ k $$
I know that 
$$\tag{2}\sum_{n\geq 1} H^{(k)}_n x^n= \frac{\operatorname{Li}_k(x)}{1-x} $$
But notice in (1) the fixed $n$.

Let $\psi(x)=\frac{\Gamma'}{\Gamma}(x)=\frac{d}{dx}\log\Gamma(x)$ be the digamma function. For $N$ a positive integer, we have
$$
\psi(x+N)-\psi(x)=\sum_{j=0}^{N-1}\frac{1}{x+j}
$$
(this follows from $x\Gamma(x)=\Gamma(x+1)$ and induction).
Now
\begin{eqnarray*}
\sum_{k\geq 1}H_n^{(k)}x^k&=&\sum_{k\geq 1}\sum_{j=1}^n \frac{1}{j^k}x^k\\
&=&\sum_{j=1}^n \sum_{k\geq 1} \left(\frac{x}{j}\right)^k\\
&=&\sum_{j=1}^n \frac{x}{j-x}\\
&=&x\sum_{j=1}^n \frac{1}{j-x}\\
&=&x\sum_{j=0}^{n-1}\frac{1}{-x+1+j}\\
&=&x(\psi(-x+1+n)-\psi(-x+1))
\end{eqnarray*}
