Hankel determinants via Stieltjes matrices (Q2718343)

The Hankel matrix \(H = (h_{ij})_{i,j \geq 0}\) is said to be generated by the integer sequence \(\langle a_i \rangle_{i \geq 0}\) if \(h_{ij} = a_{i+j}, i,j \geq 0 (a_0 := 1).\) A positive definite \(H\) admits the decomposition \(H = LDU\) where \(L\) is lower triangular with all-one main diagonal, \(U = L^T,\) and \(D\) is diagonal with \(\det (D) \not= 0.\) The Stieltjes matrix \(S_L\) associated with \(L\) is defined by \( S_L = L^{-1} \overline{L},\) where \(\overline{L}\) is obtained from \(L\) by deleting its first row. NEWLINENEWLINENEWLINELet \(H_n = (h_{ij})_{0 \leq i,j \leq n-1}\) be the Hankel matrix of order \(n\) associated with \(\langle a_i \rangle_{i \geq 0}.\) The author uses \(S_L\) to compute the sequence \(\langle \det (H_n) \rangle_{n \geq 1}\) for several classes of sequences \(\langle a_i \rangle_{i \geq 0}\) (Catalan numbers, central binomial coefficients, small and large Schröder numbers, Delannoy numbers, Motzkin numbers, directed animals numbers, Fine numbers, central trinomial coefficients, Bell numbers, derangement numbers, telephone numbers, preferential arrangements numbers) as well as to characterize some of these.