binary matrices with zero diagonal & no zero rows or columns

[Edwin Clark]

Let a(n) be the number of n x n matrices with entries in {0,1} with no
zero row and no zero column, AND with zero (main) diagonal. 

I get 

a(1) = 0, a(2) = 1, a(3) = 18, a(4) = 1699.

I find it hard to believe that this is not in the OEIS, so perhaps this is
wrong. I would appreciate it if someone would check these numbers. 

a(n) is also the number of simple labeled digraphs on n nodes for which
every vertex has indegree at least one and outdegree at least one. This
question was just raised on sci.math.research (if I interpret the question
correctly). 

Edwin