  For small $r$ the Hankel determinants $d_r(n)$ of the sequence
$\left({2n+r\choose n}\right)_{n\ge 0}$ are easy to guess and show an
interesting modular pattern. For arbitrary $r$ and $n$ no closed formulae are
known, but for each positive integer $r$ the special values $d_r(rn)$,
$d_r(rn+1)$, and $d_r(rn+\lfloor\frac{r+1}{2}\rfloor)$ have nice values which
will be proved in this paper.
