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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1202.1616","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-08T07:49:04Z","cross_cats_sorted":[],"title_canon_sha256":"8b20400c8d5d6a1248d3da22ce179012d6e569cdbc1e639be0387c003952f8b3","abstract_canon_sha256":"0909c2f6dceadfcee692e64da61d0e077437d5f79d91b24911672d2e9444c5e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:51.825591Z","signature_b64":"iQvorHQ2DjejR2K03K6pUpuNjSteVtD/hOS/FNHyuHVe7SMTxWZ16amAW2mxcSlflNAYk6jNiOgjlvXg5RMcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a64de5b50f8c8ebc5b284d82e751ca7675268f9ac334eb37bb37f4be051b70a","last_reissued_at":"2026-05-18T04:02:51.825034Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:51.825034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1202.1616","created_at":"2026-05-18T04:02:51.825117+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.1616v1","created_at":"2026-05-18T04:02:51.825117+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1616","created_at":"2026-05-18T04:02:51.825117+00:00"},{"alias_kind":"pith_short_12","alias_value":"BJSN4W2Q7DEO","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"BJSN4W2Q7DEOXRNS","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"BJSN4W2Q","created_at":"2026-05-18T12:27:01.376967+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5","json":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5.json","graph_json":"https://pith.science/api/pith-number/BJSN4W2Q7DEOXRNSQTMC45I4U5/graph.json","events_json":"https://pith.science/api/pith-number/BJSN4W2Q7DEOXRNSQTMC45I4U5/events.json","paper":"https://pith.science/paper/BJSN4W2Q"},"agent_actions":{"view_html":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5","download_json":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5.json","view_paper":"https://pith.science/paper/BJSN4W2Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.1616&json=true","fetch_graph":"https://pith.science/api/pith-number/BJSN4W2Q7DEOXRNSQTMC45I4U5/graph.json","fetch_events":"https://pith.science/api/pith-number/BJSN4W2Q7DEOXRNSQTMC45I4U5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5/action/storage_attestation","attest_author":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5/action/author_attestation","sign_citation":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5/action/citation_signature","submit_replication":"https://pith.science/pith/BJSN4W2Q7DEOXRNSQTMC45I4U5/action/replication_record"}},"created_at":"2026-05-18T04:02:51.825117+00:00","updated_at":"2026-05-18T04:02:51.825117+00:00"}