Solve recurrence relation using generating function I'm trying to solve: $a_{n+1}-a_n=n^2$, $n\le0$ , $a_0=1$ using generating functions. 
Step 1) Multiply by $x^{n+1}$
$$a_{n+1}x^{n+1}-a_nx^{n+1}=n^2x^{n+1}$$
Step 2) Take the infinite sums
$$\sum_{n\ge0}^{\infty}a_{n+1}x^{n+1}-\sum_{n\ge0}^{\infty}a_nx^{n+1}=\sum_{n\ge0}^{\infty}n^2x^{n+1}$$
Our prof gave us the identity: $$\sum_{n\ge0}^{\infty}n^2x^n= \frac{x+x^2}{{1-x}^3}$$
So I factored out an $x^1$ from my RHS(to use the identity) and simplified the LHS to get:
$$(f(x)-a_0)-xf(x)=x\left(\frac{x+x^2}{{1-x}^3}\right)$$
therefore: $$f(x)=x\left(\frac{x+x^2}{{(1-x)}^4}\right)+\frac{1}{(1-x)}=\frac{x(x+x^2)+(1-x)^3}{(1-x)^4}$$
Step 3) Decompose the function by partial fractions
Saving you all the gory details I got:
$$f(x)=\frac{4}{(1-x)^2}-\frac{5}{(1-x)^3}+\frac{2}{(1-x)^4}$$
Step 4) Finding the coefficient of $x^n$ in each term:
I recognized that each term was a  derivative of the the power series $\frac{1}{(1-x)}$ to get:
$$4[x^n]\sum(n+1)(x^n)-5\frac12[x^n]\sum(n+2)(n+1)(x^n)+2\frac13[x^n]\sum(n+3)(n+2)(n+1)(x^n)$$
So that whole thing = $a_n$ which equals:
$$4(n+1)-5\frac12(n+2)(n+1)+2\frac13(n+3)(n+2)(n+1)$$
However the answer is: $a_n=1+[n(n-1)(2n-1)\frac16]$
I tried multiplying out to see if they were the same but they're not(I checked on wolfram too).
Could anyone tell me where I went wrong?
Thank you.

$\begin{array}\\
A(x)
&=\sum_{n=0}^{\infty} a_n x^n\\
&=a_0+\sum_{n=1}^{\infty} a_n x^n\\
&=a_0+\sum_{n=0}^{\infty} a_{n+1} x^{n+1}\\
&=a_0+\sum_{n=0}^{\infty} (a_{n}+n^2) x^{n+1}\\
&=a_0+x\sum_{n=0}^{\infty} (a_{n}+n^2) x^{n}\\
&=a_0+x\left(\sum_{n=0}^{\infty} a_{n}x^{n}+\sum_{n=0}^{\infty} n^2 x^{n}\right)\\
&=a_0+x\left(A(x)+ \frac{x+x^2}{{1-x}^3}\right)\\
\end{array}
$
Now solve for
$A(x)
$.
