Can we find generating function for $\sum_{k=0}^\infty \sum_{i=0}^n(-1)^i\binom{n}{i}\binom{rk-mi-1}{n-1}$?

Question

For $n,m,r\in\mathbb{N}$,

let $A_n(m,r)$ be number of $n$-tuples, $(x_1,x_2,...,x_n)\in\mathbb{Z}^n$

such that $1\leq x_1,x_2,...,x_n\leq m$ and $r$ divides $x_1+x_2+\cdots+x_n$

I can find that $$ A_n(m,r)=\sum_{k=0}^\infty \sum_{i=0}^n(-1)^i\binom{n}{i}\binom{rk-mi-1}{n-1}$$

Now I want to find generating function $\displaystyle\sum_{n=1}^\infty A_n(m,r)x^n$

Thank for your help.