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

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Z.-W. Sun introduced three kinds of numbers: align*Sₙ=∑ₖ₌₀ⁿn k²2k k(2k+1), sₙ=∑ₖ₌₀ⁿn k²2k k1/2k-1, align* and Sₙ⁺=∑ₖ₌₀ⁿn k²2k k(2k+1)². In this paper we mainly prove that align* 4∑ₖ₌₀ⁿ⁻¹kSₖ≡ ∑ₖ₌₀ⁿ⁻¹sₖ≡ ∑ₖ₌₀ⁿ⁻¹Sₖ⁺≡ 0n² n 1, align* by establishing some binomial coefficient identities, such as align* 4∑ₖ₌₀ⁿ⁻¹kSₖ=n²∑ₖ₌₀ⁿ⁻¹1/k+12k k(6kn-1 k²+n-1 kn-1 k+1). align* This confirms several recent conjectures of Z.-W. Sun.