Conjectures on Somos $4$, $6$ and $8$ sequences using Riordan arrays and the Catalan numbers

TL;DR

This paper explores conjectures about specific families of integer sequences related to Somos sequences, using Riordan arrays and Catalan numbers, and investigates their Hankel transforms through generalized continued fractions.

Contribution

It introduces conjectural forms of Somos sequences derived from Riordan arrays and Catalan numbers, connecting combinatorial structures with Hankel transforms.

Findings

Conjectured forms of Somos 4, 6, and 8 sequences based on parameters.

Representation of sequences via stretched Riordan arrays applied to Catalan numbers.

Analysis of Hankel transforms linked to generalized Jacobi continued fractions.

Abstract

We give conjectures on the form of families of integer sequences whose Hankel transforms are, respectively, $(\alpha, \beta)$ Somos $4$ sequences, $(\alpha, 0, \gamma)$ Somos $6$ sequences, and $(\alpha, \beta, \gamma, \delta)$ Somos $8$ sequences, for particular values of $\alpha$, $\beta$, $\gamma$, $\delta$ which we describe. The sequences involved can be described in terms of the application of certain stretched Riordan arrays to the Catalan numbers, accompanied by a (sequence) Hankel transform. The combination of Riordan array and the Catalan numbers results from the study of certain generalized Jacobi continued fractions, based on the Counting Automata Methodology.