If $a_k$  is the coefficient of $x^k$  in the expansion of ${\left(1+x+x^2\right)}^n$  for $k=0,1,2,\cdots 2n$  , then $a_1+2a_2+3a_3+\cdots +2n.a_{2n}=$

Differentiating both sides with respect to $x$

$n{\left(1+x+x^2\right)}^{n-1}\left(1+2x\right)=a_1+2a_{2}x+3a_3{x}^2+\cdots +2n.a_{2n}{x}^{2n-1}$

$\text{put }x=1$

$n{\left(3\right)}^{n-1}\left(3\right)=a_1+2a_2+3a_3+\cdots +2n.a_{2n}$

$n{.3}^n=a_1+2a_2+3a_3+\cdots +2n{a}_{2n}$