What functions $f_n(k)$ have the property $$\sum_{k=r}^n k(k - 1)\ldots(k - r + 1)f_n(k) = n!\tag 1$$
I found this problem here (p. 140), so I know that one of the functions is $$f_n(k) = P_n(k) = \binom{n}{k}\sum_{i=0}^{n-k} (-1)^i\binom{n-k}{i}(n - k - i)!$$ which is a number of permutations of a set of $n$ elements which have exactly $k$ fixed points.
I have not been able to find any other functions though. I have tried to get a known identity which involves a summation on one side and a factorial on the other and rewrite it in such a way that it would resemble (1) but without success. What else can be done?
Only an idea, to be verified:
Put $\displaystyle S_n(x)=\sum_{k=0} f_n(k)x^k$. Then your hypothesis is $S_n^{(r)}(1)=n!$ for $r=0,\cdots,n$. Hence $\displaystyle S_n(x)=n!\sum_{r=0}^n \frac{(x-1)^r}{r!}$.