The coefficient of $x^n$ in the polynomial $(x+{}^n{\text{C}}_0)(x+3.{}^n{\text{C}}_1)(x+5{}^n{\text{C}}_2)\cdots [x+(2n+1){}^n{\text{C}}_n]$  is

$(x+{}^n{\text{C}}_0)(x+3.{}^n{\text{C}}_1)(x+5{}^n{\text{C}}_2)\cdots [x+(2n+1){}^n{\text{C}}_n]$

$=x^{n+1}+({}^n{\text{C}}_0+3.{}^n{\text{C}}_1+5.{}^n{\text{C}}_2+\cdots +(2n+1).{}^n{\text{C}}_n)x^n+\cdots$

$\therefore x^n\text{coefficient}=[{}^n{\text{C}}_0+3.{}^n{\text{C}}_1+5.{}^n{\text{C}}_2+\cdots +(2n+1).{}^n{\text{C}}_n]$

We know that

$a.C_0+(a+d)C_1+\cdots (a+nd)C_n=(2a+nd)2^{n-1}$

$\text{here\hspace{0.17em}\hspace{0.17em}}a=1,d=2$

$\therefore [2\left(1\right)+n\left(2\right)]2^{n-1}$

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

$=(n+1)2^n$