The asymptotic number of integer stochastic matrices

Let H^n"r be the number of n x n matrices, with nonnegative integer elements, all of whose row and column sums are equal to some prescribed integer r. Similarly, let A^n"r be the number of n x n (0.1) matrices with common row and column sum r. An asymptotic formula for H^n"r is stated and proved, the method of proof being essentially elementary. A simple modification of the proof yields an analogous asymptotic formula for A^n"r. The latter agrees with a result of O'Neil, obtained by a completely different method.