{"paper":{"title":"Inequalities for binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zhi-Hong Sun","submitted_at":"2013-10-01T15:42:45Z","abstract_excerpt":"In this paper we prove several inequalities for binomial coefficients. For instance, if $ k$ and $n$ are positive integers such that $n\\ge 400$ and $[\\frac n5]\\le k\\le [\\frac n2]$, where $[x]$ is the greatest integer not exceeding $x$, then $$\\binom nk<\\Big(1-\\frac{5(k-[\\f n5])}{6n^2}\\Big) \\frac{n^{n-\\f 12}}{k^k(n-k)^{n-k}}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0353","kind":"arxiv","version":2},"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"}