I am trying to compute the following limit (k is a fixed constant): $ \lim_{n\to\infty} \frac{ {n/2 - 1\choose(k-1)/2} {n/2 \choose (k-1)/2} }{n-1 \choose k-1} $
I expanded the binomial coefficient but I got stuck and couldn't get anywhere from there. In theory, if my approach is correct, this should converge to a constant relative to k.