Sum of coefficients of $x^i$ (Multinomial theorem application) A polynomial in $x$ is defined by
  $$a_0+a_1x+a_2x^2+ \cdots + a_{2n}x^{2n}=(x+2x^2+ \cdots +nx^n)^2.$$
  Show that the sum of all $a_i$, for $i\in\{n+1,n+2, \ldots , 2n\}$, is
  $$ \frac {n(n+1)(5n^2+5n+2)} {24}.$$


I don't know how to proceed. I know the Multinomial theorem, however, I have problems in applying it. Any help will be appreciated as it will help me understand the theorem well.

Thanks!
Here is an easy method using multinomial coefficients. Put $x=1$ to get the sum of all the coefficients. Now, we want to evaluate $\sum_{i=0}^na_i$, then we will subtract that from sum of all coefficients. Observe that these coefficients will remain unaltered even in the following expansion (because the additional terms do not contribute to powers less than $x^{n+1}$):
$$(x+2x^2+3x^3+...)^2 = x^2(1+2x+3x^2+...)^2$$
$$ = x^2\Bigg(\frac{1}{(1-x)^2}\Bigg)^2$$
$$ = \frac{x^2}{(1-x)^4}$$
$$ = x^2\sum_{m=0}^\infty\binom{m+4-1}{4-1}x^m$$
Now, apply the identity that $$\sum_{i=k}^n\binom{i}{k} = \binom{n+1}{k+1}$$
and you are done.

Hope it helps:)
You can check in the link it's similar to what your question is. 
https://www.mathsdiscussion.com/forum/topic/sum-of-coefficients/?part=1#postid-55