ag.algebraic geometry – Asymptotics of degree of \$textrm{SO}_n\$?

(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 $$N(n)$$?