I'm having a little trouble understanding this type of question:
Find the generating function for this sequence.
$$ 0,0,1,0,16,32,64,128,256,512...$$
I'm pretty new to this concept of generating functions and I don't completely understand how to solve these types of questions. Could someone point me in the right direction? Thanks.
The generating function is $$0\cdot x^0+0\cdot x^1+1\cdot x^2+0\cdot x^3+16\cdot x^4+32\cdot x^5+64\cdot x^6+\cdots.$$
The part from $16\cdot x^4$ on is $$16x^4(1+2x+4x^2+8x^3+\cdots).$$ We recognize the infinite sum as a geometric series with first term $1$ and common ratio $2x$, so sum $\frac{1}{1-2x}$ if $|x|\lt 1/2$.
Now you have all the information needed to write down a closed form for the generating function.