I don't understand how in this famous book they obtained hypergeometric coefficients for $$ \sum_{k\leq n} z^k \binom{n-k}{k}.\tag{5.74} $$
They say it is $\displaystyle F{-n,\ 1+2\lceil \frac{n}{2}\rceil\choose -1/2}$ with constant $\displaystyle\frac{z}{4}$. How did they get this ceiling? It's on page 213.
I tried and got something completely different. We have $\sum_{k\leq n}$$ n-k\choose k$, and first we transform it to $\sum_{k>0}$$ k\choose n-k$ to have "infinite" sum. So division of two consecutive terms gives $k+1\choose n-k-1$ $/$ $k\choose n-k$ - obviously something different.
What do I get wrong?