Binomial Coefficient of Minus Half

Theorem
Let $k \in \Z$.


 * $\dbinom {-\frac 1 2} k = \dfrac {\paren {-1}^k} {4^k} \dbinom {2 k} k$

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

Proof
From Product of r Choose k with r Minus Half Choose k:


 * $\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$

Setting $r = -\dfrac 1 2$: