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: {{begin-eqn}} {{eqn | l = \dbinom r {k + 1} \dbinom {r - \frac 1 2} {k + 1}       | r = \dfrac 1 {\left({r + 1}\right) \Beta \left({k + 2, r - k}\right) \left({r + \frac 1 2}\right) \Beta \left({k + 2, r - k - \frac 1 2}\right)}       | c =  }} {{eqn | r = \dfrac 1 {\left({r + 1}\right) \frac {k + 1} {r + 1} \Beta \left({k + 1, r - k}\right) \left({r + \frac 1 2}\right) \frac {k + 1} {r + \frac 1 2} \Beta \left({k + 1, r - k - \frac 1 2}\right)}       | c = Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$ }} {{eqn | r = \dfrac 1 {\left({r + 1}\right) \frac {k + 1} {r + 1} \frac {r + 1} {r - k} \Beta \left({k + 1, r - k + 1}\right) \left({r + \frac 1 2}\right) \frac {k + 1} {r + \frac 1 2} \frac {r + \frac 1 2} {r - k - \frac 1 2} \Beta \left({k + 1, r - k + \frac 1 2}\right)}       | c = Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$ }} {{eqn | r = \dfrac {\left({r - k}\right) \left({r - k  - \frac 1 2}\right)} {\left({k + 1}\right)^2} \times \frac 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)}       | c = simplifying }} {{eqn | n = 2       | r = \dfrac {\left({r - k}\right) \left({r - k  - \frac 1 2}\right)} {\left({k + 1}\right)^2} \dbinom r k \dbinom {r - \frac 1 2} k       | c = from $(1)$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \dbinom {2 r} {2 k + 2} \dbinom {2 k + 2} {2 k + 2}       | r = \dfrac 1 {\left({2 r + 1}\right) \Beta \left({2 k + 3, 2 r - 2 k - 1}\right) \left({2 k + 3}\right) \Beta \left({k + 2, k + 2}\right)}       | c =  }} {{eqn | r = \dfrac 1 {\left({2 r + 1}\right) \frac {2 k + 2} {2 r + 1} \Beta \left({2 k + 2, 2 r - 2 k - 1}\right) \left({2 k + 3}\right) \frac {k + 1} {2 k + 3} \Beta \left({k + 1, k + 2}\right)}       | c = Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$ }} {{eqn | r = \dfrac 1 {\left({2 k + 2}\right) \Beta \left({2 k + 2, 2 r - 2 k - 1}\right) \left({k + 1}\right) \Beta \left({k + 1, k + 2}\right)}       | c = simplification }} {{eqn | r = \dfrac 1 {\left({2 k + 2}\right) \frac {2 k + 1} {2 r} \Beta \left({2 k + 1, 2 r - 2 k - 1}\right) \left({k + 1}\right) \frac {k + 1} {2 k + 2} \Beta \left({k + 1, k + 1}\right)}       | c = Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$ }} {{eqn | r = \dfrac {2 r} {\left({k + 1}\right)^2 \Beta \left({2 k + 1, 2 r - 2 k - 1}\right) \left({2 k + 1}\right) \Beta \left({k + 1, k + 1}\right)}       | c = simplifying }} {{eqn | r = \dfrac {2 r} {\left({k + 1}\right)^2 \Beta \left({2 k + 1, 2 r - 2 k - 1}\right)} \binom {2 k} k       | c = Binomial Coefficient expressed using Beta Function }} {{eqn | r = \dfrac {2 r} {\left({k + 1}\right)^2 \frac {2 r} {2 r - 2 k - 1} \Beta \left({2 k + 1, 2 r - 2 k}\right)} \binom {2 k} k       | c = Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$ }} {{eqn | r = \dfrac {2 r - 2 k - 1} {\left({k + 1}\right)^2 \Beta \left({2 k + 1, 2 r - 2 k}\right)} \binom {2 k} k       | c = simplifying }} {{eqn | r = \dfrac {2 r - 2 k - 1} {\left({k + 1}\right)^2 \frac {2 r + 1} {2 r - 2 k} \Beta \left({2 k + 1, 2 r - 2 k + 1}\right)} \binom {2 k} k       | c = Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$ }} {{eqn | r = \dfrac {\left({2 r - 2 k}\right) \left({2 r - 2 k - 1}\right)} {\left({k + 1}\right)^2 \left({2 r + 1}\right) \Beta \left({2 k + 1, 2 r - 2 k + 1}\right)} \binom {2 k} k       | c = simplifying }} {{eqn | r = \dfrac {\left({2 r - 2 k}\right) \left({2 r - 2 k - 1}\right)} {\left({k + 1}\right)^2} \dbinom {2 r} {2 k} \binom {2 k} k       | c = Binomial Coefficient expressed using Beta Function }} {{eqn | r = \dfrac {4 \left({r - k}\right) \left({r - k - \frac 1 2}\right)} {\left({k + 1}\right)^2} \dbinom {2 r} {2 k} \binom {2 k} k       | c =  }} {{end-eqn}} {{finish}}