Hankel determinants for some common lattice paths

For a fixed positive integer ℓ, let f(n,ℓ) denote the number of lattice paths that use the steps (1,1), (1,−1), and (ℓ,0), that run from (0,0) to (n,0), and that never run below the horizontal axis. Equivalently, f(n,ℓ) satisfies the quadratic functional equation F(x)=∑n⩾0f(n,ℓ)xn=1+xℓF(x)+x2F(x)2. Let Hn denote the n by n Hankel matrix, defined so that (Hn)i,j=f(i+j−2,ℓ). Here we investigate the values of their determinants where ℓ=1,2,3. For ℓ=1,2 we are able to employ the Gessel-Viennot-Lindström method. For the case ℓ=3, the sequence of determinants forms a sequence of period 14, namely,(det(Hn))n⩾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.