  Given $z\in C^n$ and $A\in Z^{m\times n}$, we consider the problem of
evaluating the counting function $h(y;z):=\sum\{z^x : x\in Z^n; Ax=y, x\geq
0\}$. We provide an explicit expression for $h(y;z)$ as well as an algorithm
with possibly numerous but very simple calculations. In addition, we exhibit
finitely many fixed convex cones, explicitly and exclusively defined by $A$,
such that for any $y\in Z^m$, the sum $h(y;z)$ can be obtained by a simple
formula involving the evaluation of $\sum z^x$ over the integral points of
those cones only. At last, we also provide an alternative (and different)
formula from a decomposition of the generating function into simpler rational
fractions, easy to invert.
