  We present a multivariate generating function for all n x n nonnegative
integral matrices with all row and column sums equal to a positive integer t,
the so called semi-magic squares. As a consequence we obtain formulas for all
coefficients of the Ehrhart polynomial of the polytope B_n of n x n
doubly-stochastic matrices, also known as the Birkhoff polytope. In particular
we derive formulas for the volumes of B_n and any of its faces.
