In a few more words, if we look at the space of all nxn matrices (over a field of characteristic 0) with only 1 or 0 as an element in them ("binary matrices"), how many of them are invertible for each n? There is a short sequence on the OEIS, and from what I can tell by reading the papers referred on that page, it appears that everyone seems to be finding the (bounds of the) probability that a certain nxn matrix is invertible out of all the {1,0} matrices of same dimension.
My question is whether or not there is a way of figuring out an explicit formula that would give the total number of invertible matrices given some size n? Maybe there is some explicit proof that such a formula cannot be achieved? Or is it simply considered out of reach by modern machinery?
I also understand that brute-forcing the problem is nigh-impossible, so empirical evidence is going to be scant for a long time. Though empirical evidence isn't going to be particularly useful for a proof anyway...