Product of r Choose k with r Minus Half Choose k

Theorem

Formulation 1

Let $k \in \Z$, $r \in \R$.

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

where $\dbinom r k$ denotes a binomial coefficient.

Formulation 2

Let $k \in \Z$, $r \in \R$.

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

where $\dbinom r k$ denotes a binomial coefficient.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $47$