Binomial Coefficient of Half/Corollary

Theorem
Let $k \in \Z_{\ge 0}$.


 * $\dbinom {\frac 1 2} k = \dfrac {\paren {-1}^{k - 1} } {2^{2 k - 1} \paren {2 k - 1} } \dbinom {2 k - 1} k - \delta_{k 0}$

where:
 * $\dbinom {\frac 1 2} k$ denotes a binomial coefficient
 * $\delta_{k 0}$ denotes the Kronecker delta.

Proof
When $k > 0$ we have:

When $k = 0$ we have:

while:

Hence:


 * $(1): \quad \dbinom {\frac 1 2} k = \dfrac {2 \paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dbinom {2 k - 1} k - \delta_{k 0}$