  For a single value of $\ell$, let $f(n,\ell)$ denote the number of lattice
paths that use the steps $(1,1)$, $(1,-1)$, and $(\ell,0)$, that run from
$(0,0)$ to $(n,0)$, and that never run below the horizontal axis. Equivalently,
$f(n,\ell)$ satisfies the quadratic functional equation $F(x) = \sum_{n\ge
0}f(n,\ell) x^n = 1+x^{\ell}F(x)+x^2F(x)^2.$
  Let $H_n$ denote the $n$ by $n$ Hankel matrix, defined so that $[H_n]_{i,j} =
f(i+j-2,\ell)$. Here we investigate the values of such determinants where $\ell
= 0,1,2,3$. For $\ell = 0,1,2$ we are able to employ the
Gessel-Viennot-Lindstr\"om method. For the case $\ell=3$, the sequence of
determinants forms a sequence of period 14, namely, $$ (\det(H_n))_{n \ge 1} =
(1,1,0,0,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,0,0,-1,-1,-1,...)$$ For this case we are
able to use the continued fractions method recently introduced by Gessel and
Xin.
  We also apply this technique to evaluate Hankel determinants for other
generating functions satisfying a certain type of quadratic functional
equation.
