Number of monomials of deg D where each variables has low degree

What does "tractable mean"? You are looking for the coefficient of $x^D$ in

$$\left(\sum_{i=0}^d x^i\right)^n = \left(\frac{1-x^{d+1}}{1-x}\right)^n,$$ I believe the RHS gives something like the formula you wrote down...

This formula has no closed form. That means it can not be written as a sum of fixed number of hypergeometric terms when $d$ is greater than $2$. The proof can be found in the book A=B by Petkovsek, Wilf and Zeilberger.