Modified Balls in Boxes ; allowi "negative number of balls"

[WWGD]

Hi all,

There is a known formula for the number of solutions to

$x_1+x_2+...+x_k =n$ when $x_1,x_2,...x_n$are non-negative Integers.

Question:

Are there known formulas for the sum $x_1+x_2+...+x_k =n$

when $x_1, x_2,..,x_k$

are positive or negative Integers in a bounded range $-\infty < m \leq x_i \leq M < \infty$ (redundant) ?

Thanks.

 

[WWGD]

Never mind, thanks, just add a constant to each $x_i$ to make each term non-negative and then we refer to usual formula. Please feel free to delete; I asked a very simple/obvious question.