Compute - Coefficient Extraction

Question

I need to compute the nth coefficient from the following expression.

$[z^n]\frac{z+z^2}{\sqrt[3]{1-2z}}$

How can I proceed with this?

Answer 1

Hint:

Using Binomial series for $|-2z|<1,$

$$(1-2z)^{-1/3}=1+\sum_{r=1}^\infty\dfrac{1\cdot4\cdot(3r-1)}{3^r}(2z)^r=1+\sum_{r=1}^\infty1\cdot4\cdot(3r-1)\left(\dfrac23\right)^rz^r$$

So, $z^n, n\ge3$ in $$(z+z^2)(1-2z)^{-1/3}$$ will be $$\left(1\cdot4\cdot(3r-1)\left(\dfrac23\right)^r\right)_{r=n-1}+\left(1\cdot4\cdot(3r-1)\left(\dfrac23\right)^r\right)_{r=n-2}$$