<s>[INST] Find sequence for given generating function $\frac{ \frac{3x}{2} + \frac{3}{2}}{ 3 - x }$ I have generating function
$$A(x) = \frac{ \frac{3x}{2} + \frac{3}{2}}{ 3 - x }$$
and I need to find a sequence from it.
This was my approach:
$$ \frac{ \frac{3x}{2} + \frac{3}{2}}{ 3 - x } = \frac{-3}{2} + 2 \cdot \frac{1}{1 - \frac{x}{3}} $$
$ \frac{1}{1 - \frac{x}{3}} $ is generating function for $a_n = \frac{1}{3^n}$ so I though that the whole sequence $(c_n)_0^\infty$ would be:
$$c_n =  \frac{-3}{2} + \frac{2}{3^n} $$
but according to wolfram it is wrong, where I am doing a mistake?
 [/INST] Write $$\frac{3}{6}\frac{x}{1 - \frac{x}{3}} + \frac{3}{6} \frac{1}{1 - \frac{x}{3}}$$
and use that $$\frac{1}{1 - \frac{x}{3}} = \sum_{n=0}^{\infty} \bigg( \frac{x}{3}\bigg)^n$$
</s>