Sum of multinomial coefficients with bounded indices

Question

I want to know if there's a closed formula for $$ f_{k}(n_{1}, \dots, n_{k}) = \sum_{i_{1} = 0}^{n_{1}} \cdots \sum_{i_{k} = 0}^{n_{k}} \frac{(i_{1} + \cdots + i_{k})!}{i_{1}!\cdots i_{k}!} $$ for $k\geq 1$ and nonnegative integers $n_{1}, \dots, n_{k}$. It isn't that hard to show that $$ f_{1}(n_{1}) = n_{1} + 1 $$ and $$ f_{2}(n_{1}, n_{2}) = \binom{n_{1} + n_{2} + 2}{n_{1} + 1} - 1 $$ but I can't find any closed form formula for general $k$. Can anyone help?

Answer 1

Not an answer, just too long for a comment. For $k>1$, using the "coefficient-of" notation, we have $$f_k(n_1,\ldots n_k)=[z_1^{n_1}\cdots z_k^{n_k}]\prod_{j=1}^{k}\frac{(z_1+\ldots+z_k)^{n_j+1}-z_j^{n_j+1}}{(z_1+\ldots+z_k)-z_j}.$$ This is proved in various ways, say by using the multivariate Cauchy integral formula $$\frac{(i_1+\ldots+i_k)!}{{i_1!}\cdots{i_k!}}=\frac{1}{(2\pi\mathrm{i})^k}\oint\ldots\oint\frac{(z_1+\ldots+z_k)^{i_1+\ldots+i_k}}{z_1^{i_1+1}\cdots z_k^{i_k+1}}\,dz_1\cdots dz_k.$$ This makes the $k=2$ case almost obvious, but the denominators go complicated when $k>2$.

Also it's fairly easy to find the generating function $$\sum_{n_1,\ldots,n_k=0}^\infty f_k(n_1,\ldots,n_k)\ z_1^{n_1}\cdots z_k^{n_k}=\frac{1}{(1-z_1)\cdots(1-z_k)(1-z_1-\ldots-z_k)}.$$