TITLE: Number of monomials of deg D where each variables has low degree
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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.