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:
 * $\forall z \notin \set {-\dfrac n 2: n \in \N}: \map \Gamma z \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$

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

Hence: