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given a set of [itex]n+1 [/itex] integers [itex]a_{1},a_{2},...a_{n},\rho [/itex], is it possible to get the closed form of the generating function

[itex]f\left(x_{1},...x_{n}\right)=\sum_{i_{1},..i_{n}=0}^{\infty}\delta_{i_{1}a_{1}+...+i_{n}a_{n},\rho}x_{1}^{i_{1}}...x_{n}^{i_{n}}[/itex]

It seems to be always a rational function. For instance here are two specific results:

[itex]\sum_{i,j,k,h=0}^{\infty}\delta_{i+j-h-l,0}x_{1}^{i}x_{2}^{j}x_{3}^{k}x_{4}^{h}=\frac{1-x_{1}x_{2}x_{3}x_{4}}{\left(x_{1}x_{3}-1\right)\left(x_{2}x_{3}-1\right)\left(x_{1}x_{4}-1\right)\left(x_{2}x_{4}-1\right)}[/itex]

[itex]\sum_{i,j,k=0}^{\infty}\delta_{2i+j-h,0}x_{1}^{i}x_{2}^{j}x_{3}^{k}=\frac{1}{\left(x_{2}x_{3}-1\right)\left(x_{1}x_{3}^{2}-1\right)} [/itex]

Also references that could help me to tackle this problem would be appreciated. Thank you.

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# Generating functions

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