Some algebraic and combinatorial interpretations of lower triangular matrices from the Hankelization of sequences

Given a sequence a0, a1, a2...of real numbers, we are interested in its combinatorial interpretations. We then consider extending these combinatorial interpretations to matrices. To obtain the desired matrix, we use the Hankelization of the sequence a0, a1, a2..., where a0 = 1. The Hankel matrix is an infinite matrix H = (hn,k)n,k ≥ 0 such that hn,k = a n+k. If H is positive definite, then H = LDU where L is a lower triangular matrix with all ones on the main diagonal, U = LT, and D is a diagonal matrix. The first column of L is 1, a1, a2,.... In many cases, we find a generating function for each column of L and provide combinatorial interpretations of the columns of L as well. When we find L via Hankelization, we will name the Triangle after the sequence. For instance, if we find the LDU factorization of the Motzkin sequence via Hankelization, we will call L the Motzkin Triangle.; We give combinatorial interpretations of the Motzkin Triangle in terms of trees and paths. We then use the Stieltjes matrix of a given Riordan matrix M, with ordinary or exponential generating functions to color the edges in a tree or steps in a path. This coloring provides combinatorial interpretations for M. Non-Riordan matrices with exponential generating functions are also discussed. A class of non-Riordan matrices are obtained. We also consider the determinants of Hankel matrices of sequences with ordinary generating functions.