<s>[INST] Find generating function for the sequence 0, 0, 0, 0, 3, 4, 5, 6, ... 
1.Derive the generating function for the sequence $$0, 0, 0, 0, 3, 4, 5, 6, . . .$$
2.Derive the generating function for the sequence $$0, 0, −12, 36, −108, 324, .. .$$

So the first function looks like $3x^4 + 4x^5+5x^6...$  =  $x^4(3+4x+5x^2...)$ and that looks like the generating function for sequence $0,1,2,3,...$
 I assume the answer for the first one is something like $\frac{x^4()}{1-x}$
, what am I missing?
For the second one  if I factor $12$ out, ($-x^2+3x^3-9x^4...)$, is it something like $\frac{12x^2}{1+3x}$
 [/INST] For $|x|<1$ we obtain:
$$3x^4+4x^5+...=x^2(3x^2+4x^3+...)=x^2(x^3+x^4+...)'=$$
$$=x^2\cdot\left(\frac{x^3}{1-x}\right)'=\frac{x^4(3-2x)}{(1-x)^2}.$$
</s>