Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$.

My question:What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, that satisfy $AA^\top-A^\top A=0$?

If we restrict the attention to the subclass of symmetric matrices, my question becomes quite trivial. However, the extension to the whole class of normal matrices seems quite involved. In particular, I would like to know if this problem has already been studied in the literature.

**EDIT.** OEIS has a page dedicated to this problem, as pointed out by @Wojowu in a comment. However, only a (trivial) lower bound (in terms of the number of symmetric matrices) is listed.