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Sun introduced three kinds of numbers: Sn =∑k=0n nk2 2k k (2k+1),sn =∑k=0n nk2 2k k 1 2k − 1, and Sn+ =∑ k=0n nk2 2k k (2k+1)2. In this paper, we mainly prove that 4∑k=0n−1kS k ≡∑k=0n−1s k ≡∑k=0n−1S k+ ≡ 0(modn2)forn ≥ 1, by establishing some binomial coefficient identities, such as 4∑k=0n−1kS k = n2∑ k=0n−1 1 k + 1 2k k 6k n − 1 k2 + n − 1 k n − 1 k + 1 . This confirms several recent conjectures of Sun.

Proof of some conjectures of Z.-W. Sun on the divisibility of certain double sums