summation - Can we generalise this double sum identity?

In one of my old (Dutch) algebra/analysis problem books, I came across the following "cute" double sum identity:

$$
\sum_{k=1}^n\sum_{h=1}^k \frac{h^2-3h+1}{h!} = - n - 1 + \frac{1}{n!},

$$

which the reader was asked to prove true for all $n\in\mathbb{N}$. The proof is relatively straightforward, with induction on $n$. (Although I'd love to see a more imaginative proof!)

My question is, can we somehow generalise this identity? Because this is a rather non-obvious result, I have a hunch that the textbook writer derived it from some more general theorem. But I have no idea what to look for.

Any insights will be warmly appreciated!