{"paper":{"title":"Hankel determinants of sums of consecutive weighted Schr\\\"{o}der numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pei-Lan Yen, Sen-Peng Eu, Tsai-Lien Wong","submitted_at":"2012-02-08T07:49:04Z","abstract_excerpt":"For a real number $t$, let $r_\\ell(t)$ be the total weight of all $t$-large Schr\\\"{o}der paths of length $\\ell$, and $s_\\ell(t)$ be the total weight of all $t$-small Schr\\\"{o}der paths of length $\\ell$. For constants $\\alpha, \\beta$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $\\det_{1\\le i,j\\le n} (\\alpha r_{i+j-2}(t) +\\beta r_{i+j-1}(t))$,\n  $\\det_{1\\le i,j\\le n} (\\alpha r_{i+j-1}(t) +\\beta r_{i+j}(t))$,\n  $\\det_{1\\le i,j\\le n} (\\alpha s_{i+j-2}(t) +\\beta s_{i+j-1}(t))$, and $\\det_{1\\le i,j\\le n} (\\alpha s_{i+j-1}(t) +\\beta s_{i+j}(t))$ combinator"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1616","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"}