On the number of nonnegative solutions of a system of linear Diophantine equations

We derive a closed expression for the number of nonnegative solutions of a certain system of linear Diophantine equations. The motivation comes from high energy physics where the nonnegative solutions play a crucial role in the perturbative calculation for a class of Lagrangians describing the interaction of an atom with a boson field or a non-linear interaction of boson fields among themselves (the so-called interacting phi^n models). The linear system can be solved and the nonnegative solutions enumerated but a closed expression for the number of solutions is preferable to counting the solutions. Interestingly, the problem led to a construction of a simpler linear Diophantine system whose nonnegative number of solutions turns out to be the magic constant.