  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.
