What is the closed-form expression for
$${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$
According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the general sum of $\sum_k {\ n-k \choose k}z^k$ leads to the closed form of the above series.
I understand $\sum_k {\ n-k \choose k}z^k= \frac{1}{\sqrt{1+4z}}((\frac{1+\sqrt{1+4z}}{2})^{n+1}-(\frac{1-\sqrt{1+4z}}{2})^{n+1})$, but I can not see how this sum leads to the closed-form of the the hypergeometric series.