Sum of the coefficients of $x^{2r},r=1,2,3,\cdots$  in ${\left(1+x\right)}^n$  is

$\underset{\_}{{}^n{C}_0-{}^n{C}_{1}1x+{}^n{C}_2{x}^2-{}^n{C}_3{x}^3={\left(1-x\right)}^n}$

$2{}^n{C}_0+2{}^n{C}_2.x^2+2{}^n{C}_4.x^4={\left(1+x\right)}^n+{\left(1-x\right)}^n$

$2{}^n{C}_2+2{}^n{C}_4\cdots =\left(2^n-2\right)$

$2\left({}^n{C}_2+{}^n{C}_4+\cdots \right)=\frac{2^n-2}{2}$

$=2^{n-1}-1$