Binomial Coefficient of Real Number with Half

Theorem

 * $\dbinom r {1 / 2} = \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}$

where $\dbinom r {1 / 2}$ denotes a binomial coefficient.

Proof
Legendre's Duplication Formula gives:
 * $\forall z \notin \left\{{-\dfrac n 2: n \in \N}\right\}: \Gamma \left({z}\right) \Gamma \left (z + \dfrac 1 2 \right) = 2^{1 - 2 z} \sqrt \pi \ \Gamma \left({2 z}\right)$

and so:
 * $(1): \quad \Gamma \left ({\rho + \dfrac 1 2}\right) = \dfrac {2^{1 - 2 \rho} \sqrt \pi \ \Gamma \left({2 \rho}\right)} {\Gamma \left({\rho}\right)}$

Hence: