Is there a nice bijective proof of the fact that the determinant of the $(n+1)$-by-$(n+1)$ Hankel matrix whose respective entries are the central binomial coefficients $0 \choose 0$, $2 \choose 1$, $\dots$, $4n \choose 2n$ is $2^n$ (e.g., via Lindstrom-Gessel-Viennot)?
My first thought was to apply LGV with lattice-paths joining sources $(0,0),(1,1),\dots,(n,n)$ to sinks $(n,n),(n+1,n+1),\dots,(2n,2n)$, but one of the hypotheses of LGV isn’t satisfied.
As Ira Gessel points out in his comment, the result follows easily via a weighted version of the standard LGV proof one uses for Hankel determinants of Catalan numbers. Continued fractions aren’t needed; at a purely combinatorial level, Gessel’s weighting scheme ensures that each excursion has multiplicative weight 2, and the product of these 2’s corresponds to the multiplicity of the map that turns lattice paths in a square into Dyck paths in a triangle by folding along the diagonal joining $(0,0)$ to $(2n,2n)$.
