  The number of non-negative integer matrices with given row and column sums
appears in a variety of problems in mathematics and statistics but no
closed-form expression for it is known, so we rely on approximations of various
kinds. Here we describe a new such approximation, motivated by consideration of
the statistics of matrices with non-integer numbers of columns. This estimate
can be evaluated in time linear in the size of the matrix and returns results
of accuracy as good as or better than existing linear-time approximations
across a wide range of settings. We also use this new estimate as the starting
point for an improved numerical method for either counting or sampling matrices
using sequential importance sampling. Code implementing our methods is
provided.
