{"paper":{"title":"On some divisibility properties of binomial sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Brian Y. Sun","submitted_at":"2016-11-04T12:58:21Z","abstract_excerpt":"In this paper, we consider two particular binomial sums \\begin{align*} \\sum_{k=0}^{n-1}(20k^2+8k+1){\\binom{2k}{k}}^5 (-4096)^{n-k-1} \\end{align*} and \\begin{align*} \\sum_{k=0}^{n-1}(120k^2+34k+3){\\binom{2k}{k}}^4\\binom{4k}{2k} 65536^{n-k-1}, \\end{align*} which are inspired by two series for $\\frac{1}{\\pi^2}$ obtained by Guillera. We consider their divisibility properties and prove that they are divisible by $2n^2 \\binom{2n}{n}^2$ for all integer $n\\geq 2$. These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by $2n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01358","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"}