The coefficients of $\frac{1}{x}$ in the expansion of ${(1+x)}^n{\left(1+\frac{{\displaystyle 1}}{{\displaystyle x}}\right)}^n$ is 

${(1+x)}^n{\left(1+\frac{{\displaystyle 1}}{{\displaystyle x}}\right)}^n=\frac{{(1+x)}^{2n}}{x^n}$

$\Rightarrow {(1+x)}^n{\left(\frac{{\displaystyle x+1}}{{\displaystyle x}}\right)}^n$

$\therefore coefficentof{x}^{-1}in{(1+x)}^n{(1+\frac{1}{x})}^n$

$=coefficentof{x}^{n-1}in{(1+x)}^{2n}{=}^{2n}{C}_{n-1}$

$=\frac{2n!}{(n-1)!(2n-n+1)!}$

$=\frac{2n!}{(n-1)!(n+1)!}$