{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:P47WYCLF6REP6TKKE7F3GLZRVN","short_pith_number":"pith:P47WYCLF","schema_version":"1.0","canonical_sha256":"7f3f6c0965f448ff4d4a27cbb32f31ab51e1b4ddbeedbfecf1f87a7389a18ade","source":{"kind":"arxiv","id":"1911.05456","version":10},"attestation_state":"computed","paper":{"title":"New series for powers of $\\pi$ and related congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2019-11-13T17:30:12Z","abstract_excerpt":"Via symbolic computation we deduce 97 new type series for powers of $\\pi$ related to Ramanujan-type series. Here are three typical examples: $$\\sum_{k=0}^\\infty \\frac{P(k) \\binom{2k}k\\binom{3k}k \\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}}\n  =\\frac{18\\times557403^3\\sqrt{10005}}{5\\pi}$$ with \\begin{align*}P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\\\&+ 19850391655004126179, \\end{align*} $$\\sum_{k=1}^\\infty \\frac{(3k+1)16^k}{(2k+1)^2k^3\\binom{2k}k^3} = \\frac{\\pi^2-8}2,$$ and $$\\sum_{n=0}^\\infty\\frac{3n+1}{(-100)^n} \\sum_{k=0}^n{n\\choose k}^2T_k(1,25)T_{n-k}(1,25) = \\frac{25}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1911.05456","kind":"arxiv","version":10},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-11-13T17:30:12Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d586a1f1613a2b9cf4f2465c64b265ebeaa12316c7c0ee2dafc1ed1bec47181f","abstract_canon_sha256":"b7888a38871d77d532751a28376c34b33064de6d765dc9c706cea380e713246d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:19:37.187835Z","signature_b64":"B5ydzEK+BE+0UiKbHPtVbxG+9v/1iH7vwLjDdcj2BCcj3SuBoNy7uJDMdcAdOdC7wHlGnh7FRdOU0hiEABJmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f3f6c0965f448ff4d4a27cbb32f31ab51e1b4ddbeedbfecf1f87a7389a18ade","last_reissued_at":"2026-07-05T01:19:37.187371Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:19:37.187371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New series for powers of $\\pi$ and related congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2019-11-13T17:30:12Z","abstract_excerpt":"Via symbolic computation we deduce 97 new type series for powers of $\\pi$ related to Ramanujan-type series. Here are three typical examples: $$\\sum_{k=0}^\\infty \\frac{P(k) \\binom{2k}k\\binom{3k}k \\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}}\n  =\\frac{18\\times557403^3\\sqrt{10005}}{5\\pi}$$ with \\begin{align*}P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\\\&+ 19850391655004126179, \\end{align*} $$\\sum_{k=1}^\\infty \\frac{(3k+1)16^k}{(2k+1)^2k^3\\binom{2k}k^3} = \\frac{\\pi^2-8}2,$$ and $$\\sum_{n=0}^\\infty\\frac{3n+1}{(-100)^n} \\sum_{k=0}^n{n\\choose k}^2T_k(1,25)T_{n-k}(1,25) = \\frac{25}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.05456","kind":"arxiv","version":10},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1911.05456/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1911.05456","created_at":"2026-07-05T01:19:37.187430+00:00"},{"alias_kind":"arxiv_version","alias_value":"1911.05456v10","created_at":"2026-07-05T01:19:37.187430+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.05456","created_at":"2026-07-05T01:19:37.187430+00:00"},{"alias_kind":"pith_short_12","alias_value":"P47WYCLF6REP","created_at":"2026-07-05T01:19:37.187430+00:00"},{"alias_kind":"pith_short_16","alias_value":"P47WYCLF6REP6TKK","created_at":"2026-07-05T01:19:37.187430+00:00"},{"alias_kind":"pith_short_8","alias_value":"P47WYCLF","created_at":"2026-07-05T01:19:37.187430+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/P47WYCLF6REP6TKKE7F3GLZRVN","json":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN.json","graph_json":"https://pith.science/api/pith-number/P47WYCLF6REP6TKKE7F3GLZRVN/graph.json","events_json":"https://pith.science/api/pith-number/P47WYCLF6REP6TKKE7F3GLZRVN/events.json","paper":"https://pith.science/paper/P47WYCLF"},"agent_actions":{"view_html":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN","download_json":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN.json","view_paper":"https://pith.science/paper/P47WYCLF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1911.05456&json=true","fetch_graph":"https://pith.science/api/pith-number/P47WYCLF6REP6TKKE7F3GLZRVN/graph.json","fetch_events":"https://pith.science/api/pith-number/P47WYCLF6REP6TKKE7F3GLZRVN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN/action/storage_attestation","attest_author":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN/action/author_attestation","sign_citation":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN/action/citation_signature","submit_replication":"https://pith.science/pith/P47WYCLF6REP6TKKE7F3GLZRVN/action/replication_record"}},"created_at":"2026-07-05T01:19:37.187430+00:00","updated_at":"2026-07-05T01:19:37.187430+00:00"}