{"paper":{"title":"Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2014-12-10T00:38:55Z","abstract_excerpt":"Z.-W. Sun introduced three kinds of numbers: \\begin{align*}S_n=\\sum_{k=0}^{n}{n\\choose k}^2{2k\\choose k}(2k+1),\\qquad s_n=\\sum_{k=0}^{n}{n\\choose k}^2{2k\\choose k}\\frac{1}{2k-1}, \\end{align*} and $S_n^{+}=\\sum_{k=0}^{n}{n\\choose k}^2{2k\\choose k}(2k+1)^2$. In this paper we mainly prove that \\begin{align*} 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 $n\\geqslant 1$}, \\end{align*} by establishing some binomial coefficient identities, such as \\begin{align*} 4\\sum_{k=0}^{n-1}kS_k=n^2\\sum_{k=0}^{n-1}\\frac{1}{k+1}{2k\\choose k}(6k{n-1\\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5415","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}