Binomial Coefficient of Half

Theorem

Let $k \in \Z$.

$\dbinom {\frac 1 2} k = \dfrac {\left({-1}\right)^{k - 1} } {4^k \left({2 k - 1}\right)} \dbinom {2 k} k$

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

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.

$\ds \dbinom {\frac 1 2} k$

$=$

$\ds \dfrac {1/2} {1/2 - k} \dbinom {1/2 - 1} k$

$\ds $

$=$

$\ds \dfrac 1 {1 - 2 k} \dbinom {-1/2} k$

$\ds $

$=$

$\ds \dfrac 1 {1 - 2 k} \dfrac {\left({-1}\right)^k} {4^k} \dbinom {2 k} k$

$\ds $

$=$

$\ds \dfrac {\left({-1}\right)^{k - 1} } {4^k \left({2 k - 1}\right)} \dbinom {2 k} k$

simplifying

$\blacksquare$