The analytic properties of Hoggatt triangles

Abstract

The $d$-Hoggatt triangle is a lower triangular matrix whose entries are given by specific minors of Pascal's triangle formed by consecutive $d$ rows and $d$ columns. The cases $d=1,2,3$ correspond to Pascal's triangle, the Narayana triangle, and the Baxter triangle, respectively. In this paper, we present the infinite log-concavity of the row and column sequences, the log-concavity of the sequences along transversals, and the eventual log-convexity of the sequences along rays of the $d$-Hoggatt triangle. In addition, we prove the asymptotic normality of the row sequences and total positivity of the $d$-Hoggatt triangle.