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 $prod_{j=0}^{n-1}(2^n-2^j)$.