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I'm wondering if there's a special polynomial with a name out there with $x_1,x_2,\ldots,x_k$ as variables that's defined like this:

$$ \sum_{\substack{i_1>0,i_2>0, \ldots,i_k>0 \\ i_1 +i_2+\cdots+i_k=n}} \frac{n!}{i_1! i_2! \cdots i_k!} x_1^{i_1} x_2^{i_2}\cdots x_k^{i_k} $$

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This is the multinomial formula, or multinomial theorem, a generalization of Newton's binomial formula. It is the expansion of $(x_1 + x_2 + ... + x_k)^n$. – Manolito Pérez Mar 11 '12 at 7:14
@ManolitoPérez I think you guys didn't notice that $i_1>0,i_2>0,...,i_k>0$ – Tianyang Li Mar 11 '12 at 7:27
@BrianM.Scott I think you guys didn't notice that $i_1>0,i_2>0,...,i_k>0 $ – Tianyang Li Mar 11 '12 at 7:28
My apologies: indeed I did not, despite the title. – Brian M. Scott Mar 11 '12 at 8:06
This form might help: $$\frac{n!}{(n-k)!}\int_0^{x_1}\cdots\int_0^{x_k} (u_1+\cdots+u_k)^{n-k}du_1\cdots du_k,$$ which appears to also be: $$\sum_{J\subseteq[k]}(-1)^{k-\#J}\left(\sum_{j\in J} x_j\right)^n.$$ – anon Mar 11 '12 at 8:48

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