Consider all $n\times n$ binary (entries are either $0$ or $1$) matrices, denoted $\mathcal{B}_n$. Define the $X$-ray sequence of $A=(a_{ij})\in\mathcal{B}$ 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 $n!(n+1)!$.
Example. Let $A=\begin{pmatrix} 1&2&3\\3&4&5\\0&1&2\end{pmatrix}$. Then $X(A)=15762$.
QUESTION. If we specialize to the subfamily $\mathcal{F}_n\subset\mathcal{B}_n$ of such invertible (over the field $\mathbb{F}_2$) matrices, then is there a formula for the total number $u_n$ 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 $\prod_{j=0}^{n-1}(2^n-2^j)$.
UPDATE. I've now computed a few terms: $u_1=1, u_2=5, u_3=77, u_4=2150$.