From Summation from k to m of r Choose k by s Choose n-k by nr-(r+s)k: :$\displaystyle \sum_{k \mathop = 0}^m \dbinom r k \dbinom s {n - k} \paren {n r - \paren {r + s} k} = \paren {m + 1} \paren {n - m} \dbinom r {m + 1} \dbinom s {n - m}$ Set $r - \dfrac 1 2$ and $s = -\dfrac 1 2$. :$\displaystyle \sum_{k \mathop = 0}^m \dbinom {1/2} k \dbinom {-1/2} {n - k} \paren {\frac n 2 - \paren {\frac 1 2 - \frac 1 2} k} = \paren {m + 1} \paren {n - m} \dbinom {1/2} {m + 1} \dbinom {-1/2} {n - m}$ Take the {{LHS}}: {{begin-eqn}} {{eqn | o =        | r = \sum_{k \mathop = 0}^m \dbinom {1/2} k \dbinom {-1/2} {n - k} \paren {\frac n 2 - \paren {\frac 1 2 - \frac 1 2} k}       | c =  }} {{eqn | r = \dfrac n 2 \sum_{k \mathop = 0}^m \dbinom {1/2} k \dbinom {-1/2} {n - k}       | c = immediate simplification }} {{eqn | r = \dfrac n 2 \sum_{k \mathop = 0}^m \dbinom {1/2} k \dfrac {\left({-1}\right)^{n - k} } {4^{n - k} } \dbinom {2 n - 2 k} {n - k}       | c = Binomial Coefficient of Minus Half }} {{eqn | r = \dfrac n 2 \sum_{k \mathop = 0}^m \paren {\dfrac {\left({-1}\right)^{k - 1} } {2^{2 k - 1} \left({2 k - 1}\right)} \dbinom {2 k - 1} k - \delta_{k 0} } \dfrac {\left({-1}\right)^{n - k} } {2^{2 n - 2 k} } \dbinom {2 n - 2 k} {n - k}       | c = Binomial Coefficient of Half: Corollary }} {{eqn | r = \dfrac n 2 \sum_{k \mathop = 0}^m \dfrac {\left({-1}\right)^{n - 1} } {2^{2 n - 1} \left({2 k - 1}\right)} \dbinom {2 k - 1} k \dbinom {2 n - 2 k} {n - k} - \dfrac n 2 \dfrac {\left({-1}\right)^n} {2^{2 n} } \dbinom {2 n} n       | c = separating out the $\delta_{k 0}$ }} {{eqn | r = \dfrac {\left({-1}\right)^n n} {2^{2 n} } \paren {\sum_{k \mathop = 0}^m \dbinom {2 k - 1} k \dbinom {2 n - 2 k} {n - k} \dfrac {-1} {\left({2 k - 1}\right)} - \dfrac 1 2 \dbinom {2 n} n}       | c = simplifying }} {{end-eqn}} Now the {{RHS}}: {{begin-eqn}} {{eqn | o =        | r = \paren {m + 1} \paren {n - m} \dbinom {1/2} {m + 1} \dbinom {-1/2} {n - m}       | c =  }} {{eqn | r = \paren {m + 1} \paren {n - m} \dbinom {1/2} {m + 1} \dfrac {\left({-1}\right)^{n - m} } {2^{2 n - 2 m} } \dbinom {2 n - 2 m} {n - m}       | c = Binomial Coefficient of Minus Half }} {{eqn | r = \paren {m + 1} \paren {n - m} \dfrac {\left({-1}\right)^m} {\left({2 m + 1}\right) 2^{2 m + 2} } \dbinom {2 m + 2} {m + 1} \dfrac {\left({-1}\right)^{n - m} } {2^{2 n - 2 m} } \dbinom {2 n - 2 m} {n - m}       | c = Binomial Coefficient of Half }} {{eqn | r = \dfrac 1 {2^{2 n + 2} } \paren {m + 1} \paren {n - m} \dfrac {\left({-1}\right)^n} {2 m + 1} \dbinom {2 m + 2} {m + 1} \dbinom {2 n - 2 m} {n - m}       | c = simplification }} {{eqn | r = \dfrac 1 {2^{2 n + 2} } \paren {m + 1} \paren {n - m} \dfrac {\left({-1}\right)^n} {2 m + 1} \dfrac {2 m + 2} {m + 1} \dbinom {2 m + 1} m \dbinom {2 n - 2 m} {n - m}       | c = Factors of Binomial Coefficient }} {{eqn | r = \dfrac {\left({-1}\right)^n} {2^{2 n + 1} } \paren {n - m} \dfrac {m + 1} {2 m + 1} \dbinom {2 m + 1} m \dbinom {2 n - 2 m} {n - m}       | c = simplification }} {{eqn | r = \dfrac {\left({-1}\right)^n} {2^{2 n + 1} } \paren {n - m} \dfrac {m + 1} {2 m + 1} \dfrac {2 m + 1} {2 m + 1 - m} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m}       | c = Factors of Binomial Coefficient: Corollary 1 }} {{eqn | r = \dfrac {\left({-1}\right)^n} {2^{2 n + 1} } \paren {n - m} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m}       | c = simplification }} {{end-eqn}} Thus we have: {{begin-eqn}} {{eqn | l = \dfrac {\left({-1}\right)^n n} {2^{2 n} } \paren {\sum_{k \mathop = 0}^m \dbinom {2 k - 1} k \dbinom {2 n - 2 k} {n - k} \dfrac {-1} {\left({2 k - 1}\right)} - \dfrac 1 2 \dbinom {2 n} n}       | r = \dfrac {\left({-1}\right)^n} {2^{2 n + 1} } \paren {n - m} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m}       | c = as the {{RHS}} equals the {{LHS}} }} {{eqn | ll= \leadsto       | l = \sum_{k \mathop = 0}^m \dbinom {2 k - 1} k \dbinom {2 n - 2 k} {n - k} \dfrac {-1} {\left({2 k - 1}\right)} - \dfrac 1 2 \dbinom {2 n} n       | r = \dfrac {n - m} {2 n} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m}       | c = dividing both sides by $\dfrac {\left({-1}\right)^n n} {2^{2 n} }$ }} {{eqn | ll= \leadsto       | l = \sum_{k \mathop = 0}^m \dbinom {2 k - 1} k \dbinom {2 n - 2 k} {n - k} \dfrac {-1} {\left({2 k - 1}\right)}       | r = \dfrac {n - m} {2 n} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m} + \dfrac 1 2 \dbinom {2 n} n       | c =  }} {{end-eqn}} {{qed}}