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,...

$\displaystyle x^3/(1-x)^2$

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

$\displaystyle (1/(1-x))-x^2$

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

$\displaystyle 2/(1+x)$

d)$\displaystyle C(7,0),C(7,1), C(7,3), \ldots, C(7,7),0,0,0,0,0,\ldots$

$\displaystyle (1+x)^7$

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

$\displaystyle 2/(1-2x)$

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

$\displaystyle x/(1-x^3)$

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

$\displaystyle x^3/(1-x)$

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

$\displaystyle 2x(1-x^6)/(1-x)$