# nt.number theory – Enumerating binary matrices by \$X\$-ray sequences

Consider all $$ntimes n$$ binary (entries are either $$0$$ or $$1$$) matrices, denoted $$mathcal{B}_n$$.
Define the $$X$$-ray sequence of $$A=(a_{ij})inmathcal{B}$$ by $$X(A)=x(1)x(2)cdots x(2n-1)$$ where
$$x(k)=sum_{i+j=k+1}a_{ij}$$. Then, the number of distinct $$X$$-ray sequences can be easily seen to be $$n!(n+1)!$$.

QUESTION. If we specialize to the subfamily $$mathcal{F}_nsubsetmathcal{B}_n$$ of invertible (over the field $$mathbb{F}_2$$) such matrices, then is there a formula for the total number of distinct $$X$$-ray sequences? If this is asking too much, how about an asymptotic growth of such enumeration?

NOTE. The cardinality of $$mathcal{F}_n$$ is $$prod_{j=0}^{n-1}(2^n-2^j)$$.