Proof of Sun's Conjecture on the Divisibility of Certain Binomial Sums

Abstract

In this paper, we prove the following result conjectured by Z.-W. Sun:
$$
(2n-1){3n\choose n}|
\sum_{k=0}^{n}{6k\choose 3k}{3k\choose k}{6(n-k)\choose 3(n-k)}{3(n-k)\choose n-k}
$$
by showing that the left-hand side divides each summand on the right-hand side.