I don't understand generating functions...can someone please shine some light on this topic?

I don't see how to get from one point to another....for example:

Find a closed form for the generating functions of each of these sequences:

Each of the following already have the solutions, I just don't understand how to get to them.

a)0,0,0,1,2,3,4,...

b)1,1,0,1,1,1,1,1,1,1,...

c)2,-2,2,-2,2,-2,2,-2,...

d)

e)2,4,8,16,32,64,128,256,...

f) 0,1,0,0,1,0,0,1,0,0,1,...

g)0,0,0,1,1,1,1,1,1,...

h)0,2,2,2,2,2,2,0,0,0,0,0...