It's late at night and I'm tired, but I just stumbled across this while doing my homework. Any chance this is new? Or, maybe, did I just somehow transform it and it is still basically the same formula? In that case, forgive me, please.
Anyway, here it is. It is a recursive function (thus, of limited use!?) to calculate the sum of factorials:
$f(n) = \sum\limits_{i=1}^n i! = \frac{(n+1)!}{n}+f(n-2)$
with
$f(0) = 0$ and $f(-1) = -1$
Is this useful at all or did I just waste my time? :)