Hankel-type determinants for some 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 Apéry 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 $n=0,1,2,\dots$, we show that $6^{-n}|f_{i+j}{|}_{0\le i,j\le n}$ and $12^{-n}|D_{i+j}{|}_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}{|}_{0\le i,j\le n}$ and $24^{-n}|A_{i+j}{|}_{0\le i,j\le n}$ are always integers.

13 pages, final published version

References in corpus (1)

• On some determinants with Legendre symbol entries