Polynomials and the IMO I have been stuck on this problem, as I am unable to proceed properly. The problem is as follows:  
If $f(x) = (x+2x^2+\cdots nx^n)^2 = a_2x^2 + a_3x^3 +\cdots a_{2n}x^{2n},$ prove that $$a_{n+1} + a_{n+2} +\cdots +a_{2n} = \binom{n+1}{2}\frac{5n^2+5n+2}{12}$$    
I tried expanding the LHS but only ended up with a bunch of expressions. Any help is appreciated.

Hint
See that:
$$a_j= \sum_{i=1}^{j}i(j-i)=\frac{j^3-j}{6}$$
And then
$$\sum_{j=n+1}^{2n}a_j=\frac{1}{6}\sum_{j=n+1}^{2n}(j^3-j)=\frac{1}{6}\left(\sum_{j=1}^{2n}(j^3-j)-\sum_{j=1}^{n}(j^3-j)\right)$$
