I can't find the pattern while evaluating $\int_0^{\pi/2}\sin^n(x)\,dx$

These are Wallis' integrals : https://en.wikipedia.org/wiki/Wallis%27_integrals

An integration by parts leads to $W_{n+2}=\frac{n+1}{n+2}W_n$ therefore $W_{2n}=\frac{(2n)!\pi}{2^{2n+1}(n!)^2}$ and $W_{2n+1}=\frac{4^n(n!)^2}{(2n+1)!}$.