On Wikipedia, https://en.wikipedia.org/wiki/Gamma_function, I read that the following hold true for any positive integer $n$:
\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{n}n!}{\sqrt {\pi }}={\frac {(2n-1)!!}{2^{n}}}{\sqrt {\pi }}={\binom {n-{\frac {1}{2}}}{n}}n!{\sqrt {\pi }}\end{aligned}
Clearly, the last identity cannot hold true, since the upper entry of the binomial coefficient is smaller than the lower entry. What would be the correct entries of the binomial coefficient in order for the identity to hold true?
To answer this question, it is not necessary to know the definition of Gamma function, since the problem only involves the last identity.