The value of  $1\cdot C_1+3\cdot C_3+5\cdot C_5+7\cdot C_7+\dots \dots$

There  $C_0,C_1,C_2,\dots \dots .C_n$ are the binomial coefficients in the  

expansion of  $(1+x{)}^n$ is 

We have, 

$\sum _{r=1}^n r{\cdot }^n{C}_r=n\cdot 2^{n-1}$ and $\sum _{r=1}^n (-1{)}^{r-1}\cdot r^n{C}_r=0$

Adding these two, we get 

 $\begin{array}{l}2\left(1\cdot C_1+3\cdot C_3+5\cdot C_5+7\cdot C_7+\dots .\right)=n\cdot 2^{n-1}\\ \Rightarrow 1\cdot C_1+3\cdot C_3+5\cdot C_5+\dots \dots ..=n\cdot 2^{n-2}\end{array}$