number of solutions for the equation

Question

What is the total number of solutions of the equation of the form $$x_1 + x_2 + \dots+ x_r \ge 0 $$ such that $-k_1 \le x_1 \le k_1; -k_2\le x_2 \le k_2; \dots -k_r\le x_r \le k_r$ where $K_i \ge 0 \forall i \in \{1,\dots ,r\}$ .

Answer 1

Let $y_i=k_i-x_i$ for $i=1,\dots,r$, and let $k=k_1+\ldots+k_r$, then you’re looking for non-negative integer solutions to $y_1+\ldots+y_r\le k$ subject to the conditions $y_i\le 2k_i$ for $i=1,\dots,r$. Now add an $(r+1)$-st variable $y_0$, and count the solutions to $y_0+y_1+\ldots+y_r=k$ in non-negative integers satisfying $y_i\le 2k_i$ for $i=1,\dots,r$.

Without the upper bounds this would be a standard stars-and-bars problem, and the answer would be $\binom{k+r}r$. With the restrictions it requires an inclusion-exclusion argument that is rather messy. My answer to this problem covers the case in which the $k_i$ are all equal.