\[\genfrac{(}{)}{0.0pt}{}{n_{1}+n_{2}+\cdots+n_{k}}{n_{1},n_{2},\ldots,n_{k}}=%
\frac{(n_{1}+n_{2}+\cdots+n_{k})!}{n_{1}!\,n_{2}!\,\cdots\,n_{k}!}=\prod_{j=1}%
^{k-1}\genfrac{(}{)}{0.0pt}{}{n_{j}+n_{j+1}+\cdots+n_{k}}{n_{j}}.\]