A "binomial" generalization of harmonic numbers
Solution 1:
Indeed (thanks to darij grinberg; writing it out just to have the question answered), $$\sum_{s=1}^{\infty}\frac{H_s(n)}{s}t^s=\sum_{k=1}^{n}\sum_{s=1}^{\infty}\frac{1}{s}\Big(\frac{t}{k}\Big)^s=-\ln\prod_{k=1}^{n}\Big(1-\frac{t}{k}\Big)$$ so the result follows from $(*)$ and "exponential formula". In turn, $(*)$ is a consequence of $$\prod_{k=1}^{n}\Big(1-\frac{t}{k}\Big)^{-1}=\sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}\Big(1-\frac{t}{k}\Big)^{-1}$$ which is simply the partial fraction expansion of the LHS.