In some notes I have the following formula:
$y_n = \frac{2^{-2n}(2n)!}{n!n!}$
Then it simplifies that to be $y_n = \frac{2^{-2n}e^{-2n}2^{2n}\sqrt{4\pi n}}{e^{-2n}2^{2n}2\pi n} = \frac{1}{\sqrt{\pi n}}$
So it applies what I was told here is the stirling approximation to the factorial i.e.
$n! \approx e^{-n}*n^n*\sqrt{2\pi n}$
But I am not sure how to fully apply it to get the end result. I mean how do we handle $(2n)!$ in this formula to simlify it as mentioned above?