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I'm trying to prove this generating function identity. Let k be a fixed positive integer.

$$ \sum_{{a_1,\ldots,}_{}{a_k\ \geq 0}}\min{(a_1,\ldots,a_k)} x_{_1}^{a_1}x_{_2}^{a_2}\cdots x_{_k}^{a_k}=\frac{x_1x_2\cdots x_k}{(1-x_1)(1-x_2)\cdots(1-x_k)(1-x_1x_2\cdots x_k)} $$

$a_1, a_2, \ldots, a_k $, denotes all positive $k$-tuples and $\min( a_1, a_2, \ldots, a_k)$ denotes the smallest number among $a_1, a_2, \ldots, a_k $.

I've never solved a multivariable generating function before. In a single variable case I would go straight to partial fractions, but it does not appear to apply in this situation.

Any help would be greatly appreciated.

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Hint: What is the coefficient of $x_1^{a_1}\cdots x_k^{a_k}$ in $\displaystyle\sum_{m\ge1}^\infty (x_1\cdots x_k)^m\sum_{e_1,\cdots,\,e_k\ge0}x_1^{e_1}\cdots x_k^{e_k}~$?

(It's the $\#$ of ways of writing $(a_1,\cdots,a_k)=(e_1+m,\cdots,e_k+m)$ with $m\ge1,e_i\ge0$, which is...)

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