Dilated Hankel determinants

For a sequence $\mathbf a=(a_0,a_1,\dots)$ we define its dilated Hankel determinant $\ddot{H}_n(\mathbf a)=\det(a_{2i+j})_{0\le i,j\le n-1}$, the minor of the infinite Hankel matrix $(a_{i+j})$ formed from the even-indexed rows and the first $n$ columns. We prove that, for a broad class of sequences, $\ddot{H}_n$ admits a remarkably simple product evaluation. This mirrors the behaviour of the classical Hankel determinant $H_n$, but with two key distinctions: the class of sequences for which such formulas are known is far larger in the classical case; and, whereas $H_n$ enjoys a single universal evaluation -- the Heilermann formula via the Jacobi continued fraction -- no analogous general method exists for the dilated determinant, which is therefore considerably more challenging. Our evaluations instead rest on six methods developed here, four of general scope and two of a more specialised nature. The cases treated include the factorial numbers, the Catalan and central binomial coefficients; the Euler numbers and a one-parameter secant family; the involution numbers; the Springer numbers along with elliptic and derivative deformations; the reciprocal-sine function, whose evaluation rests on a new Catalan determinant proved by condensation; a Bessel analogue of the Euler numbers; and a multiplicative Bessel family. As an application, we settle a conjecture of Chapoton and the author on the roots of the Poupard and Kreweras polynomials.