{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XYIWH3OG2ISVQQZROPTZIZW47R","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7bc363cec059e8eed2000e87f8ada24e2a10f398932f61eadb93314f678a80ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-22T03:57:57Z","title_canon_sha256":"99544cfa81bdab6a570ceffb03c26d5cd7f7dcbbb54a07dad5fe7b90f3881248"},"schema_version":"1.0","source":{"id":"1609.06810","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.06810","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"arxiv_version","alias_value":"1609.06810v4","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06810","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"pith_short_12","alias_value":"XYIWH3OG2ISV","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XYIWH3OG2ISVQQZR","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XYIWH3OG","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:494cf004f81ae6e414a8e673ec13541853d0ad65121614d657e346b17fcc40e6","target":"graph","created_at":"2026-05-18T00:09:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Bao-Xuan Zhu, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-22T03:57:57Z","title":"Hankel-type determinants for some combinatorial sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06810","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0928ab5bac657ac36e4a0283517d315fe5994bc1e69cb416398c3d340aa38f27","target":"record","created_at":"2026-05-18T00:09:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7bc363cec059e8eed2000e87f8ada24e2a10f398932f61eadb93314f678a80ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-22T03:57:57Z","title_canon_sha256":"99544cfa81bdab6a570ceffb03c26d5cd7f7dcbbb54a07dad5fe7b90f3881248"},"schema_version":"1.0","source":{"id":"1609.06810","kind":"arxiv","version":4}},"canonical_sha256":"be1163edc6d22558433173e79466dcfc4c6b3b1405ed413717f4d1f661360bed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be1163edc6d22558433173e79466dcfc4c6b3b1405ed413717f4d1f661360bed","first_computed_at":"2026-05-18T00:09:04.390448Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:04.390448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TvBFZCpP2MHjKYsfWkU5OR3U+T8PsZ2JiOthZ6swgCvSntgS2PTH9uYr5qOpxGSfVdmd4PW4WSgA4re7Cj2XCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:04.391038Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.06810","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0928ab5bac657ac36e4a0283517d315fe5994bc1e69cb416398c3d340aa38f27","sha256:494cf004f81ae6e414a8e673ec13541853d0ad65121614d657e346b17fcc40e6"],"state_sha256":"d2f2ca0b40ed8f2fc9d25be88ab6d25c2ad2bddcdef70e8825e0525ab21dc658"}