Generating function for $\sum_{k\geq 1} H^{(k)}_n x^ k $

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
$$\tag{2}\sum_{k\geq 1} H^{(k)}_n x^n= \frac{\operatorname{Li}_k(x)}{1-x} $$
But notice in (1) the fixed $n$.