A Refinement of Guo’s Theorem Concerning Divisibility Properties of Binomial Coefficients

Abstract

Let $s(n,k)=(\genfrac{}{}{0ex}{}{6k}{3k})(\genfrac{}{}{0ex}{}{3k}{k})((\genfrac{}{}{0ex}{}{3(n-k)}{n-k})/(2n-1)(\genfrac{}{}{0ex}{}{3n}{n}))$. Recently, Guo confirmed a conjecture of $Z.-W$. Sun by showing that $s(n,k)$ is an integer for $k=0,1,\dots ,n$. Let $d=(3n+2)/gcd(3n+2,2n–1)$. In this paper, we prove that $s(n,k)$ is a multiple of the odd part of $d$ for $k=0,1,\dots ,n$. Furthermore, if $gcd(k,n)=1$, then $s(n,k)$ is also a multiple of $n$. We also show that the $2$-adic order of $s(n,k)$ is at least the sum of the digits in the binary expansion of $3n$.