  Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative
integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i =
sum_{j=1}^n t_j. Let M(\svec,\tvec) be the number of m*n matrices with
nonnegative integer entries such that the i-th row has row sum s_i and the j-th
column has column sum t_j for all i,j. Such matrices occur in many different
settings, an important example being the contingency tables (also called
frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j.
Previous work has established the asymptotic value of M(\svec,\tvec) as
m,n\to\infty with s and t bounded (various authors independently, 1971-1974),
and when \svec,\tvec are constant vectors with m/n,n/m,s/n >= c/log n for
sufficiently large (Canfield and McKay, 2007). In this paper we extend the
sparse range to the case st=o(S^(2/3)). The proof in part follows a previous
asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill,
McKay and Wang, 2006). We also generalise the enumeration to matrices over any
subset of the nonnegative integers that includes 0 and 1.
