Express $a_n=\frac{1.5.9.13...(4n+1)}{2^n}$ in terms of gamma function

Question

$a_n=\frac{1.5.9.13...(4n+1)}{2^n}$

I've tried albeit unsuccessfully to generate gamma functions that will be equal to the above.

Tried using the gamma function of fractions but didn't achieve anything with it:

$$\Gamma(\frac{2n-1}{2})=[\frac{(2n-3)(2n-5)...1}{2^{n-1}}]\pi$$

Who can help?

Answer 1

Try $\Gamma({13\over4})$, or generally, $\Gamma(n+{1\over4})$. The problem is, you will end up with $\Gamma({1\over4})$ and won't be able to get rid of it. That's how things work.