Greatest coefficient in the expansion ${(x+\frac{1}{x})}^{2n+1}$ is

In the expansion n is odd, so greatest binomial coefficient is ${}^n{\text{C}}_{\frac{n-1}{2}}={}^{2n+1}{\text{C}}_{\frac{2n+1-1}{2}}$

$={}^{2n+1}{\text{C}}_n$

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