The number of integral points in a polytope
Check out

https://www.math.ucdavis.edu/~latte/

and the corresponding paper

Effective lattice point counting in rational convex polytopes. Jesús A. De Loera, Raymond Hemmecke, Jeremiah Tauzer, Ruriko Yoshida. Journal of Symbolic Computation 2004 38:4, 1273-1302.

The algorithm they implement was introduced in 

A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed. Alexander I. Barvinok. Mathematics of Operations Research 1994 19:4, 769-779.