An interesting class of Hankel determinants

TL;DR

This paper investigates Hankel determinants of a binomial coefficient sequence, revealing modular patterns and special value formulas for specific indices, advancing understanding of their structure.

Contribution

It provides new explicit formulas for certain Hankel determinants of binomial sequences at specific indices, extending known results.

Findings

Identifies modular patterns in small r cases

Derives explicit formulas for d_r(rn), d_r(rn+1), and d_r(rn+floor((r+1)/2))

Establishes properties of Hankel determinants for the sequence

Abstract

For small $r$ the Hankel determinants $d_r(n)$ of the sequence $\left({2n+r\choose n}\right)_{n\ge 0}$ are easy to guess and show an interesting modular pattern. For arbitrary $r$ and $n$ no closed formulae are known, but for each positive integer $r$ the special values $d_r(rn)$, $d_r(rn+1)$, and $d_r(rn+\lfloor\frac{r+1}{2}\rfloor)$ have nice values which will be proved in this paper.