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

Let . Let be a tuple of positive integers such that . Let such that .

Given the equation and the conditions . If I wanted to calculate the number of integer solutions to this Diophantine equation, I would start with the following:

This is the number of integer solutions with , 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:

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 , then we can write this as:

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