What is the number of $n\times n$ 0/1matrices with rank $k$? (The rank is taken over the rationals.)
This sequence is OEIS A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n) According to comments rows add to $2^{n^2}$. There are some references and pari/gp code. 

