{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:3DUCOVSCW4SI2W3Y4VOBDDJP34","short_pith_number":"pith:3DUCOVSC","schema_version":"1.0","canonical_sha256":"d8e8275642b7248d5b78e55c118d2fdf14523e8620c60e986a25ef4b15370118","source":{"kind":"arxiv","id":"1008.2894","version":3},"attestation_state":"computed","paper":{"title":"New congruences for sums involving Apery numbers or central Delannoy numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2010-08-17T13:41:05Z","abstract_excerpt":"The Ap\\'ery numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\\sum_{k=0}^{n}{n+k\\choose 2k}^2{2k\\choose k}^2, \\quad D_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences:\n\\sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\\equiv \\sum_{k=0}^{n-1}\\varepsilon^k (2k+1)^{2r+1}D_k \\equiv 0\\pmod n,\nwhere $n\\geqslant 1$, $r\\geqslant 0$, and $\\varepsilon=\\pm1$. For $r=1$, we further show that\n\\sum_{k=0}^{n-1}(2k+1)^{3}A_k &\\equiv 0\\pmod{n^3}, \\quad\n\\sum_{k=0}^{p-1}(2k+1)^{3}A_k &\\equiv p^3 \\pmod{2p^6},\nwhere $p>3$ is a prime"},"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":"1008.2894","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-17T13:41:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"40f887f481943428638ae0b12c8dc5ac2ed77e353b3dc9d8fcb0c6776980f568","abstract_canon_sha256":"ca244957db69265539ca3fb61f3665eb8c3367e7b1991e89c32205a25fd189ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:00.602824Z","signature_b64":"3oibgDMJiW05XWPDi1J2xEpgynZAIL+mF5ZuNzG6eN9o2uIt5kJfOuHgqEBDmCB2x+narf+VaYQpsJ1cV+0qBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8e8275642b7248d5b78e55c118d2fdf14523e8620c60e986a25ef4b15370118","last_reissued_at":"2026-05-18T03:55:00.602229Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:00.602229Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New congruences for sums involving Apery numbers or central Delannoy numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2010-08-17T13:41:05Z","abstract_excerpt":"The Ap\\'ery numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\\sum_{k=0}^{n}{n+k\\choose 2k}^2{2k\\choose k}^2, \\quad D_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences:\n\\sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\\equiv \\sum_{k=0}^{n-1}\\varepsilon^k (2k+1)^{2r+1}D_k \\equiv 0\\pmod n,\nwhere $n\\geqslant 1$, $r\\geqslant 0$, and $\\varepsilon=\\pm1$. For $r=1$, we further show that\n\\sum_{k=0}^{n-1}(2k+1)^{3}A_k &\\equiv 0\\pmod{n^3}, \\quad\n\\sum_{k=0}^{p-1}(2k+1)^{3}A_k &\\equiv p^3 \\pmod{2p^6},\nwhere $p>3$ is a prime"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2894","kind":"arxiv","version":3},"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":"1008.2894","created_at":"2026-05-18T03:55:00.602332+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.2894v3","created_at":"2026-05-18T03:55:00.602332+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.2894","created_at":"2026-05-18T03:55:00.602332+00:00"},{"alias_kind":"pith_short_12","alias_value":"3DUCOVSCW4SI","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_16","alias_value":"3DUCOVSCW4SI2W3Y","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_8","alias_value":"3DUCOVSC","created_at":"2026-05-18T12:26:03.138858+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/3DUCOVSCW4SI2W3Y4VOBDDJP34","json":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34.json","graph_json":"https://pith.science/api/pith-number/3DUCOVSCW4SI2W3Y4VOBDDJP34/graph.json","events_json":"https://pith.science/api/pith-number/3DUCOVSCW4SI2W3Y4VOBDDJP34/events.json","paper":"https://pith.science/paper/3DUCOVSC"},"agent_actions":{"view_html":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34","download_json":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34.json","view_paper":"https://pith.science/paper/3DUCOVSC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.2894&json=true","fetch_graph":"https://pith.science/api/pith-number/3DUCOVSCW4SI2W3Y4VOBDDJP34/graph.json","fetch_events":"https://pith.science/api/pith-number/3DUCOVSCW4SI2W3Y4VOBDDJP34/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34/action/storage_attestation","attest_author":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34/action/author_attestation","sign_citation":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34/action/citation_signature","submit_replication":"https://pith.science/pith/3DUCOVSCW4SI2W3Y4VOBDDJP34/action/replication_record"}},"created_at":"2026-05-18T03:55:00.602332+00:00","updated_at":"2026-05-18T03:55:00.602332+00:00"}