Number of $n\times n$ non-singular $0-1$ matrices Problem: Find the number of $n\times n$ non-singular matrices where an entry is either $0$ or $1$.
PS: This was motivated by 779412

The answer corresponds to OEIS sequence A055165. Little seems to be known apart from the forst eight values:
The proportion of singular matrices (among the $2^{n^2}$ candidates) is conjectured to be ${n+1\choose 2}\frac 1{2^{n-1}}$ 
