Hankel-type determinants for some combinatorial sequences

TL;DR

This paper proves conjectures about Hankel-type determinants of certain combinatorial sequences, showing their divisibility properties and integrality, thus advancing understanding of their algebraic and combinatorial structure.

Contribution

It confirms several conjectures on Hankel determinants for sequences like Franel, Domb, and Apéry numbers, establishing their divisibility and integrality properties.

Findings

$6^{-n}|f_{i+j}|$ and $12^{-n}|D_{i+j}|$ are positive odd integers

$10^{-n}|b_{i+j}|$ and $24^{-n}|A_{i+j}|$ are integers

Confirmed conjectures on Hankel determinants for combinatorial sequences

Abstract

In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\'ery numbers. For any nonnegative integer $n$, define \begin{gather*}f_n:=\sum_{k=0}^n\binom nk^3,\ D_n:=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}, b_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k,\ A_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2. \end{gather*} For $n=0,1,2,\ldots$, we show that $6^{-n}|f_{i+j}|_{0\leq i,j\leq n}$ and $12^{-n}|D_{i+j}|_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\leq i,j\leq n}$ and $24^{-n}|A_{i+j}|_{0\leq i,j\leq n}$ are always integers.