Asymptotic bound for products of even binomial coefficients

Question

So I want to create a tight bound for this product of even binomial coefficients $$\prod_{k=1}^{\lfloor{(n-2)/2}\rfloor} {n-1 \choose 2k}.$$ It's clear that a crude upper bound would be $$O\big(2^{n\lfloor{(n-2)/2}\rfloor}\big)$$ since any binomial coefficient ${n \choose k} = O(2^n)$. But is there a tighter asymptotic bound I can find, so I can maybe state in terms of big theta? Are there any properties of products of binomial coefficients I could be pointed to for help?