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

${(1+\mathrm{x}+{\mathrm{x}}^2)}^{\mathrm{n}}={\mathrm{a}}_0+{\mathrm{a}}_1 \mathrm{x}+{\mathrm{a}}_2 {\mathrm{x}}^2+....{\mathrm{a}}_{2\mathrm{n}} {\mathrm{x}}^{2\mathrm{n}}$

Diff with respect to 'x' and Put x=1

$\begin{array}{rcl}\mathrm{n}{(1+\mathrm{x}+{\mathrm{x}}^2)}^{\mathrm{n}-1} (1+2\mathrm{x})& =& {\mathrm{a}}_1+2{\mathrm{a}}_2 \mathrm{x}+3{\mathrm{a}}_3 {\mathrm{x}}^2+....+2\mathrm{n} {\mathrm{a}}_{2\mathrm{n}} {\mathrm{x}}^{2\mathrm{n}-1}\\ \mathrm{n}{3}^3& =& {\mathrm{a}}_1+2{\mathrm{a}}_2+3{\mathrm{a}}_3+....+2\mathrm{n} \mathrm{a}{2}_{\mathrm{n}}\end{array}$