Bell polynomial

$$B_{n,k}(x_1,x_2,\dots,x_{n-k+1})=\sum{n! \over j_1!j_2!\cdots j_{n-k+1}!} \left({x_1\over 1!}\right)^{j_1}\left({x_2\over 2!}\right)^{j_2}\cdots\left({x_{n-k+1} \over (n-k+1)!}\right)^{j_{n-k+1}}$$ where the sum is taken over all sequences $j_1,j_2,j_3,\ldots j_{n-k+1}$ of non-negative integers such that $j_1+j_2+\ldots=k$ and $j_1+2j_2+3j_2+\ldots=n$.