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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.

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    I forgot to mention that $n$ is even and $k$ is odd, so the binomial operator still holds.2012-11-06

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Keeping only the leading terms in $n$ yields (with $\ell=k-1$)

$ \frac{(n/2)^{\ell/2}(n/2)^{\ell/2}}{n^\ell}\cdot\frac{\ell!}{(\ell/2)!^2}=2^{-\ell}\binom\ell{\ell/2}\approx\frac1{\sqrt{\pi\ell/2}}\;, $

where the estimate on the right is asymptotic for $k\to\infty$.

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    @joriki Makes sense.. Thanks a lot2012-11-06