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We describe the computer-driven \\emph{discovery and proof} of the fact that the generating series $G(x)= \\sum_{n \\geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: \\[ G(x) = 1 + 6 \\cdot \\int_0^x \\frac{\\,\\pFq21{1/3}{2/3}{2} {\\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \\, dw.\\]"},"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":"1105.4456","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SC","submitted_at":"2011-05-23T10:12:12Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"d5032c886c4a61eaece67bf5b53cd3dbc8c3aa92802e4837a5be36c171e2c14a","abstract_canon_sha256":"232c2654b4036f35a4ec0c430d70d3dd85cff8996b2220c033400c3f78f2b39a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:00.565737Z","signature_b64":"0KAsDMlEk8kv+LcdoS2HhKTtU9LxmcF0TGxdffswbKlCZq+PPgCWIViIKetAIpwNbluoZqfVVkfAYnVe+xChDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef9049e33806354018d60c0034a6895de074108b7e6a52d52f1df8928fc99483","last_reissued_at":"2026-05-18T04:12:00.565216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:00.565216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit formula for the generating series of diagonal 3D rook paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.SC","authors_text":"Alin Bostan, Fr\\'ed\\'eric Chyzak, Lucien Pech, Mark van Hoeij","submitted_at":"2011-05-23T10:12:12Z","abstract_excerpt":"Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \\times n \\times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \\emph{discovery and proof} of the fact that the generating series $G(x)= \\sum_{n \\geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: \\[ G(x) = 1 + 6 \\cdot \\int_0^x \\frac{\\,\\pFq21{1/3}{2/3}{2} {\\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \\, dw.\\]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4456","kind":"arxiv","version":2},"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":"1105.4456","created_at":"2026-05-18T04:12:00.565295+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.4456v2","created_at":"2026-05-18T04:12:00.565295+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4456","created_at":"2026-05-18T04:12:00.565295+00:00"},{"alias_kind":"pith_short_12","alias_value":"56IETYZYAY2U","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"56IETYZYAY2UAGGW","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"56IETYZY","created_at":"2026-05-18T12:26:20.644004+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/56IETYZYAY2UAGGWBQADJJUJLX","json":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX.json","graph_json":"https://pith.science/api/pith-number/56IETYZYAY2UAGGWBQADJJUJLX/graph.json","events_json":"https://pith.science/api/pith-number/56IETYZYAY2UAGGWBQADJJUJLX/events.json","paper":"https://pith.science/paper/56IETYZY"},"agent_actions":{"view_html":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX","download_json":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX.json","view_paper":"https://pith.science/paper/56IETYZY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.4456&json=true","fetch_graph":"https://pith.science/api/pith-number/56IETYZYAY2UAGGWBQADJJUJLX/graph.json","fetch_events":"https://pith.science/api/pith-number/56IETYZYAY2UAGGWBQADJJUJLX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX/action/storage_attestation","attest_author":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX/action/author_attestation","sign_citation":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX/action/citation_signature","submit_replication":"https://pith.science/pith/56IETYZYAY2UAGGWBQADJJUJLX/action/replication_record"}},"created_at":"2026-05-18T04:12:00.565295+00:00","updated_at":"2026-05-18T04:12:00.565295+00:00"}