Trinomial coefficients and a determinant of Sun

Hai-Liang Wu, Yue-Feng She, He-Xia Ni

Published 2021-08-24Version 1

In this paper, by using the tool of trinomial coefficients we study some determinant problems posed by Zhi-Wei Sun. For example, given any odd prime $p$ with $p\equiv 2\pmod 3$, we show that $2\det[\frac{1}{i^2-ij+j^2}]_{1\le i,j\le p-1}$ is a quadratic residue modulo $p$. This confirms a conjecture of Zhi-Wei Sun.