Proof of $S_{n,r} = \frac{1}{r!} \sum_{k=0}^r (-1)^{r-k} \binom{r}{k} k^n$ with inclusion-exclusion principle
Finding a closed form expression for the recurrence relation $a(n,k+1)=(k+1)^n+\sum_{m=1}^k {k+1\choose m}a(n,m)$