Let $D,n,d$ be three positive integers.


I am looking for the number of monomials of degree $D$ in $n$ variables where each variable appears with exponent at most $d$.


As a result of an application of inclusion/exclusion principle I found the following expression
\begin{equation\*}
\sum\_j (-1)^j \binom{n}{j} \binom{D-j(d+1)+(n-1)}{n-1}
\end{equation\*}
where the summation runs over a set of indices where the expression makes sense: $0\leq j\leq n$, $j \leq D/(d+1)$.


However, I am looking for a more tractable formula, if there exists one.



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...


