If  $C_r=\left(\begin{array}{l}2n\\ r\end{array}\right),$  then   ${C_1}^2-2.{C_2}^2+3.{C_3}^2-\mathrm{.......}-2n.{{{C_{2n}}^n}^{ }}_{ }=$

$C_0+C_{1}x+C_2{x}^2+C_3{x}^3+\mathrm{....}={(1+x)}^{2n}$

Differentiate to get 

$C_1+2.C_{{ }_2}x+3.C_{{ }_3}{x}^{{ }^2}+\mathrm{....}=2n{(1+x)}^{2n-1}$    further,

$C_0{x}^{2n}-C_1{x}^{2n-1}+C_2{x}^{2n-2}-\mathrm{...}={(x-1)}^{2n}$

Multiplying above two and considering the coefficients of $x^{2n-1}$ , we get the desired series as the coefficient of  $x^{2n-1}$ in

$-2n{(1+x)}^{2n-1}{(1-x)}^{2n}=2n(x-1){(1-x^2)}^{2n-1}$

Or the coefficient of  $x^{2n-2}$ is  $2n{(1-x^2)}^{2n-1}$

$=2n{(-1)}^{n-1}\left(\begin{array}{l}2n-1\\ n-1\end{array}\right)={(-1)}^{n-1}\frac{\left(2n\right)!}{n!(n-1)!}$