My question: Is there a known closed form for the following:

Let $\displaystyle m,n \in \mathbb{Z}^+$. Let $\displaystyle (a_0, a_1, \cdots, a_m) \in \mathbb{Z}^{m+1}$ be a tuple of positive integers such that $\displaystyle \displaystyle \sum_{i=0}^m a_i = n$. Let $\displaystyle k \in \mathbb{Z}$ such that $\displaystyle 0 \le k \le n$.

Given the equation $\displaystyle b_0+b_1+\cdots + b_m = k$ and the conditions $\displaystyle \forall i, 0 \le b_i \le a_i$. If I wanted to calculate the number of integer solutions to this Diophantine equation, I would start with the following:

$\displaystyle \displaystyle \binom{m+k}{m}$

This is the number of integer solutions with $\displaystyle 0\le b_i$, but the variables have no upper limit. So, I have to subtract the number of cases where the upper limit(s) is/are violated. I can calculate this using the inclusion/exclusion principle. First, I calculate the number of ways that a single variable's upper limit is violated:

$\displaystyle \displaystyle \sum_{i = 0}^m \left\{\begin{matrix}\binom{m+k-a_i}{m}, & \text{if } a_i \le k \\ 0, & \text{if } a_i > k\end{matrix}\right.$

I subtract this sum from the original binomial. Then, I add back when exactly two upper limits are violated, then subtract when exactly three upper limits are violated, etc.

If $\displaystyle [m] = \{0,1,\ldots,m\}$, then we can write this as:

$\displaystyle \displaystyle \sum_{A \subseteq [m]} \left\{\begin{matrix}(-1)^{|A|}\begin{pmatrix}m+k- \displaystyle \sum_{i \in A} a_i \\m\end{pmatrix}, & \text{if } \displaystyle \sum_{i \in A} a_i \le k \\ 0, & \text{if } \displaystyle \sum_{i \in A} a_i > k\end{matrix}\right.$

Is this the best we can do for a closed form?