I'm trying to find the closed form of the ordinary generating function for the following sequences:
(1) $2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0,\dotsc.$
(2) $0, 0, 1, 3, 9, 27, 81, 243,\dotsc$
Here is my work. I think I've figured out the sequences. My problem is, how do I get these into closed form? From class, I'm used to seeing closed form in other ways; for example, the geometric series would be $1/(1-x)$; binomial theorem would be $(1+z)^n$. But how would I write these in closed form? Also, am I correct in thinking that these numbers are meant to represent coefficients?