Proof of some conjectures of Z.-W. Sun on the divisibility of certain double sums (Q2800145)

Numerous conjectures involving the numbers NEWLINE\[NEWLINES_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}(2k+1),NEWLINE\]NEWLINE NEWLINE\[NEWLINE s_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}\frac{1}{2k-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE S_n^{+}=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}(2k+1)^2, NEWLINE\]NEWLINE proposed by \textit{Z.-W. Sun} [``Two new kinds of numbers and related divisibility results'', Preprint, \url{arXiv:1408.5381}] are here confirmed by employing the method of induction, the WZ algorithm developed by \textit{M. Petkovšek} et al. [\(A=B\). Wellesley, MA: A. K. Peters (1996; Zbl 0848.05002)] and two congruences due to \textit{H. Pan} and \textit{Z. Sun} [Discrete Math. 306, No. 16, 1921--1940 (2006; Zbl 1221.11052)] and refined by \textit{Z.-W. Sun} and \textit{R. Tauraso} [Adv. Appl. Math. 45, No. 1, 125--148 (2010; Zbl 1231.11021)].NEWLINENEWLINEIn particular, the authors prove some binomial coefficient identities in terms of the Legendre symbol and the Catalan numbers like NEWLINE\[NEWLINE 4\sum_{k=0}^{n-1}kS_k=n^2\sum_{k=0}^{n-1}\frac{1}{k+1}{2k\choose k}\left(6k{n-1\choose k}^2+{n-1\choose k}{n-1\choose k+1}\right) NEWLINE\]NEWLINE and subsequently they establish that NEWLINE\[NEWLINE 4\sum_{k=0}^{n-1}kS_k\equiv \sum_{k=0}^{n-1}s_k\equiv \sum_{k=0}^{n-1}S_k^{+}\equiv 0\pmod{n^2}\quad\text{for \(n\geq 1\)}. NEWLINE\]NEWLINENEWLINENEWLINEThe paper also suggests a way to solve a \(q\)-analogue open problem still formulated by Sun.