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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 $(x_1 + x_2 + ... + x_k)^n$.2012-03-11
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    @ManolitoPérez I think you guys didn't notice that $i_1>0,i_2>0,...,i_k>0$2012-03-11
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    @BrianM.Scott I think you guys didn't notice that $i_1>0,i_2>0,...,i_k>0 $2012-03-11
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    My apologies: indeed I did not, despite the title.2012-03-11
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    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: $$\sum_{J\subseteq[k]}(-1)^{k-\#J}\left(\sum_{j\in J} x_j\right)^n.$$2012-03-11
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    @anon Has such a function being studied before? I'm looking for some asymptotic results on this function.2012-03-11
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    @tmy1018: I wouldn't be surprised if some version is somewhere in *Enumerative Combinatorics*, but I've never seen a name or any special attention attached to it.2012-03-11
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    Sorry, I didn't notice that either.2012-03-16

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