Binomial Coefficient of Half/Corollary

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


 * $\dbinom {\frac 1 2} k = \dfrac {\left({-1}\right)^{k - 1} } {2^{2 k - 1} \left({2 k - 1}\right)} \dbinom {2 k - 1} k - \delta_{k 0}$

where $\dbinom {\frac 1 2} k$ denotes a binomial coefficient.

Proof
When $k > 0$ we have:

When $k = 0$ we have:

while:

Hence:


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

where $\delta_{k 0}$ denotes the Kronecker delta.