Find a generating function of $(0,0,1\cdot2^1,0,0,2\cdot2^2,0,0,3\cdot2^3,\ldots)$

Question

Find a generating function of: $$(0,0,1\cdot2^1,0,0,2\cdot2^2,0,0,3\cdot2^3,\ldots)$$ We can write this as a generating function: $$f(x)=x^0\cdot0+ x^1\cdot0 + x^2\cdot1\cdot2^1 + \cdots$$ Which is:: $$f(x)=x^2\cdot1\cdot2^1 + x^5\cdot2\cdot2^2 + x^8\cdot3\cdot2^3 + \cdots$$ We can: $$f(x)=x^2(1\cdot2^1 + x^3\cdot2\cdot2^2 + x^6\cdot3\cdot2^3 + \cdots)$$ But I've been sitting on this for a while and I don't know how to do this, I mean how to simplify this to get simple formula.

Answer 1

$$f(x) = x^2 \sum_{n=1}^\infty n x^{3n-3} 2^n = 2x^2 \sum_{n=1}^\infty n (2x^3)^{n-1} = \frac{2 x^2}{(1-2 x^3)^2}$$

Answer 2

HINT.-With some work I seem to have found the solution. I leave final details for those who want to give the last touch. (Some care perhaps with the first terms of the summations). $$f(x)=\sum n2^n x^{3n-1}\Rightarrow xf(x)=\sum n(2x^3)^n$$ $$\frac{1}{(1-2x^3)^2}=\sum n(2x^3)^{n-1}\Rightarrow\frac{2x^3}{(1-2x^3)^2}=\sum n(2x^3)^n$$