(This is an offshoot of Degree of parametrization of $textrm{SO}_n$?)

Consider $G=textrm{SO}_n$ as an affine subvariety of the affine space of $N$-by-$N$ matrices. There is an explicit expression for $deg(G)$ in Theorem 4.2. of https://arxiv.org/abs/1701.03200 : $deg(G) = 2^{n-1} N(n)$, where $N(n)$ is the number of non-intersecting lattice paths from the points $(n-2 i,0)$, $1leq ileq lfloor n/2rfloor$, to the points $(0,n-2 j)$, $1leq jleq lfloor n/2rfloor$. What are the asymptotics of $N(n)$?