If ${(1+x+2x^2)}^{20}=a_0+a_{1}x+a_2{x}^2+\mathrm{.....}+a_{40}{x}^{40}$ then the value of $a_1+a_3+a_5+\mathrm{....}+a_{37}$ is equal to

Put x = 1 & x = – 1 
$\begin{array}{l}{a}_0+a_1+a_2+\mathrm{......}=2^{40}\\ a_0-a_1+a_2-a_3+a_4-a_5+\mathrm{......}=2^{20}\\ 2[a_1+a_3+a_5+\mathrm{.....}+a_{37}+a_{39}]=2^{20}(2^{20}-1)\\ a_1+a_3+a_5+\mathrm{.....}+a_{37}=2^{19}(2^{20}-1)-a_{39}\\ \therefore a_1+a_3+a_5+\mathrm{.....}+a_{37}=2^{19}(2^{20}-21)\end{array}$
$a_{39}=$coefficient of $x^{39}$ in ${(1+x+2x^2)}^{20}=2^{19}.20$