Let \(s(n, k) = \binom{6k}{3k} \binom{3k}{k} (\binom{3(n-k)}{n-k} / (2n-1) \binom{3n}{n})\). Recently, Guo confirmed a conjecture of \(Z.-W\). Sun by showing that \(s(n, k)\) is an integer for \(k = 0, 1, \ldots, n\). 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 \(k = 0, 1, \ldots, n\). 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 \(3n\).