{"paper":{"title":"Hankel-type determinants for some combinatorial sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bao-Xuan Zhu, Zhi-Wei Sun","submitted_at":"2016-09-22T03:57:57Z","abstract_excerpt":"In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\\'ery numbers. For any nonnegative integer $n$, define \\begin{gather*}f_n:=\\sum_{k=0}^n\\binom nk^3,\\ D_n:=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k\\binom{2(n-k)}{n-k}, b_n:=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k,\\ A_n:=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k^2. \\end{gather*} For $n=0,1,2,\\ldots$, we show that $6^{-n}|f_{i+j}|_{0\\leq i,j\\leq n}$ and $12^{-n}|D_{i+j}|_{0\\le i,j\\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\\leq i,j\\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06810","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"}