generating function for a power sequence

Question

The question is short: I don't understand how should I solve this. Problem wants the G(x) of this: 1,4,9,16,... I can solve this one but I cannot connect these two to each other: 1,2,3,4,...

Answer 1

HINT: Since you didn’t mention $0$, you apparently want

$$G(x)=\sum_{n\ge 0}(n+1)^2x^n\;.$$

Start with

$$\frac1{1-x}=\sum_{n\ge 0}x^n$$

and differentiate with respect to $x$ to get

$$\frac1{(1-x)^2}=\sum_{n\ge 0}(n+1)x^n\;.$$

Now multiply both sides by $x$ and ...

(If you actually want $\sum_{n\ge 0}n^2x^n$, some minor adjustment is necessary, but the same idea works.)