Product of r Choose k with r Minus Half Choose k/Formulation 1/Proof 2

Proof
From Binomial Coefficient expressed using Beta Function:


 * $(1): \quad \dbinom r k \dbinom {r - \frac 1 2} k = \dfrac 1 {\left({r + 1}\right) \Beta \left({k + 1, r - k + 1}\right) \left({r + \frac 1 2}\right) \Beta \left({k + 1, r - k + \frac 1 2}\right)}$

Then:

Again from Binomial Coefficient expressed using Beta Function:
 * $(3): \quad \dbinom {2 r} k \dbinom {2 r - k} k = \dfrac 1 {\left({2 r + 1}\right) \Beta \left({k + 1, 2 r - k + 1}\right) \left({2 r - k + 1}\right) \Beta \left({k + 1, 2 r - 2 k + 1}\right)}$

Then: