Proof of Sun's conjecture on the divisibility of certain binomial sums

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.

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