$\displaystyle A_n = A ^ 2_{ n - 1} / A_{ n - 2 }$..... if $\displaystyle A_0 = 1$ and $\displaystyle A_1 = 2$.....
No need to use a generating function. Just work out the first few values of $\displaystyle A_n$ and you will soon see that $\displaystyle A_n=2^n$. Then you can easily prove that by induction.