Find a generating function for $a_r = n^3$ What is the generating function for $a_r = n^3$? I computed an answer, just wanted to double check my answer.

Here is how you advance. Assume 
$$ F(x) = \sum_{r=0}^{\infty} a_r x^r \implies F(x)=\sum_{r=0}^{\infty} r^3 x^r $$
$$ \implies F(x)= (xD)(xD)(xD)\sum_{r=0}^{\infty} x^r = (xD)^3 \frac{1}{1-x}, $$
where $D=\frac{d}{dx}$.
Can you finished now?
Added Here is the final answer

$$ F(x)={\frac {x \left( 1+4\,x+{x}^{2} \right) }{ \left( 1-x \right) ^{4}}}. $$

