we know that the number of integer solutions for the following equation with integers $m, r$:
$\begin{equation} z_1 + \cdots + z_m = 2m, \quad 1 \leq z_i \leq r \end{equation}$
is $N = \binom{2m-1}{m-1} - \binom{m}{1} \binom{2m-r-1}{m-1} + \binom{m}{2} \binom{2m-2r-1}{m-1} - \binom{m}{3} \binom{2m-3r-1}{m-1}+ \binom{m}{4} \binom{2m-4r-1}{m-1}- \cdots$
I am interesting in the asymptotic behavior of $N$ when $m$ goes to infinity. My first try is to derive a upper bound for $N$, but I do not know how to do that. Can anyone give me some hints?
