{"paper":{"title":"On monotonicity of some combinatorial sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haomin Wen, Qing-Hu Hou, Zhi-Wei Sun","submitted_at":"2012-08-19T23:30:05Z","abstract_excerpt":"We confirm Sun's conjecture that $(\\root{n+1}\\of{F_{n+1}}/\\root{n}\\of{F_n})_{n\\ge 4}$ is strictly decreasing to the limit 1, where $(F_n)_{n\\ge0}$ is the Fibonacci sequence. We also prove that the sequence $(\\root{n+1}\\of{D_{n+1}}/\\root{n}\\of{D_n})_{n\\ge3}$ is strictly decreasing with limit $1$, where $D_n$ is the $n$-th derangement number. For $m$-th order harmonic numbers $H_n^{(m)}=\\sum_{k=1}^n 1/k^m\\ (n=1,2,3,\\ldots)$, we show that $(\\root{n+1}\\of{H^{(m)}_{n+1}}/\\root{n}\\of{H^{(m)}_n})_{n\\ge3}$ is strictly increasing."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3903","kind":"arxiv","version":8},"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"}