- Put $B(n,k)=k! S(n,k)$, here $S(n,k)$ be the Srirling numbers of second kind.

Prove that $$ \sum_{j=i+1}^n \frac{(-1)^{j+1-i}}{j-i} B(n,j)= n B(n-1,i),i=0,1,\ldots,n-1 $$

- Put $T(n,i)=\displaystyle \sum_{j=i}^n (-1)^{j-i} 2^{n-j} B(n,j) {j-1 \choose i-1}$.

Prove that

$$ \sum_{j=i+1}^n \frac{1-(-1)^{j-i}}{2(j-i)}T(n,j)= n T(n-1,i),i=0 \ldots n-1. $$