{"paper":{"title":"Asymptotic estimate for the polynomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiyou Li","submitted_at":"2014-05-08T05:18:13Z","abstract_excerpt":"The polynomial coefficient $\\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\\binom {n,q}{cn}-\\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor$ or $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\\lfloor\\log_2{n}\\rfloor$ or $q=\\lfloor\\log_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1803","kind":"arxiv","version":4},"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"}