I am interested in the quantity $ a_{n} = \sqrt{n/2} \frac{\Gamma((n-1)/2)}{\Gamma(n/2)}$ (this is the geometric bias of the non-central t-distribution with $n$ d.f.) After some plotting, my hunch is that $a_n \approx 1 + \frac{3}{4n} + \mathcal{O}\left(n^{-2}\right)$, as $n\to\infty$. Is there a known asymptotic result of this form? One may assume $n$ is an integer.
A little googling lead me to Feng Qi's excellent survey of inequalities around ratios of $\Gamma$ functions of this form, but I cannot find an asymptotic expansion of this form.