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the sum $x_1+x_2+....+x_k =n$ for $x_i,n $ integers and $|x_i|<M < \infty$ has a closed form, from a question I asked ( with a trivial answer). Is there a closed form for $$x_1+x_2+...+x_k \leq n $$ with similar constraints, i.e., $ |x_i|<M< \infty, x_i \in \mathbb Z $ ?

This sum is clearly equal to $$\Sigma_{j=-M}^n C(j+k-1, k-1)$$ (where we adjust the sum by adding $M$ to each term), where $C(a,b)$ is "a choose b" = $\frac {a!}{b!(a-b)!}$ (sorry don't know how to Tex this), but I dont know of a closed for this expression. Any ideas? Thanks.

gary
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  • I do not know a closed form. But then again I do not know precisely what "closed form" means. – André Nicolas Oct 30 '15 at 15:21
  • Something that would allow me to do the sum and get a number in "not too many steps" so that I dont have to add term-by-term. – gary Oct 30 '15 at 15:27
  • @AndréNicolas: Could you please give me an idea of the answer you have? – gary Oct 30 '15 at 21:03
  • I do not know a closed form. The bounds on the $x_i$ can be dealt with by Inclusion/Exclusion. In certain cases it is feasible, but the general expressions are quite horrible. One can get a generating function, but the evaluation of coefficients is difficult. – André Nicolas Oct 30 '15 at 21:20
  • @gary: For the inclusion-exclusion formula see my answer to this question. – Brian M. Scott Oct 31 '15 at 00:07

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