Q:

Summation of multinominal coefficients with extra bounds on summation indices

My question is related to the sum 
\begin{equation}
S(n,N) = \sum_{k_1+k_2+...+k_N=n}\frac{n!}{(k_1!)\cdot(k_2!)\cdot...\cdot(k_N!)}  = N^n,
\end{equation}
which is comes from the multinomial theorem: https://en.wikipedia.org/wiki/Multinomial_theorem. 
Recently I came across the very similar expression
\begin{equation}
\tilde{S}(n,N) = \sum_{\substack{k_1+k_2+...+k_N=n\\k_1+2\cdot k_2+...+N\cdot k_N=N}}\frac{n!}{(k_1!)\cdot(k_2!)\cdot...\cdot(k_N!)}, 
\end{equation}
which has an extra restriction $\sum_{i=1}^N i\cdot k_i=N$ on the summation indices $k_i$. I'm not sure how to approach this when trying to evaluate $\tilde{S}$, so any input and answers are much appreciated! 

A:

This is a sum of coefficients of $t^N\prod x_i^{k_i}$ in $(tx_1+t^2x_2+\dots)^n$. In other words, it is the coefficient of $t^N$ in $$(t+t^2+\dots)^n=t^n (1-t)^{-n}=\sum \binom{-n}{k}(-1)^k t^{k+n},$$
hence your sum equals $$\binom{-n}{N-n}(-1)^{N-n}=\binom{N-1}{N-n}.$$

