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For example, we prove that $$\\sum_{k=0}^\\infty\\frac{(49k+1)8^k}{3^k\\binom{3k}k}=81+16\\sqrt3\\,\\pi \\ \\ \\text{and}\\ \\ \\sum_{k=0}^\\infty\\frac{10k-1}{\\binom{4k}{2k}}=\\frac{4\\sqrt 3}{27}\\pi.$$\n  We also establish the following efficient formula for computing $\\log n$ with $1<n\\le 85/4$: \\begin{align*} &\\sum_{k=0}^\\infty\\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\\binom{4k}{2k}}\\\\ \\ \\ &=6n(n+1)(n-1)^3\\log n-32n(n+1)^2(n^2-4n+1), \\end{align*} where $$P(n):=n^6-58"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2204.08275","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-11T16:39:25Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"51b96590271d1d73f1433fe00970289441143fb01c63d34908f89334e936de54","abstract_canon_sha256":"26aa4377483b3ac2416a36078656e25cfc19d95318491f1bd3e95d0d7dbedb75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:22:41.361923Z","signature_b64":"YVmh1ae0/hhtMy6T5ULrKEw3KoEUOf7eM0um4X7WSdLy1CfVvza3Y85RvgjgHUXLMTqXu98e9JB/iJqSiIVYAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5834c9064d04af65e020ccfde33ebc45e2b1f8ea5516368a948ac73e16d862b","last_reissued_at":"2026-07-05T10:22:41.361282Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:22:41.361282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evaluations of some series of the type $\\sum_{k=0}^\\infty(ak+b)x^k/\\binom{mk}{nk}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2022-04-11T16:39:25Z","abstract_excerpt":"In this paper, via the beta function we evaluate some series of the type $\\sum_{k=0}^\\infty(ak+b)x^k/\\binom{mk}{nk}$. For example, we prove that $$\\sum_{k=0}^\\infty\\frac{(49k+1)8^k}{3^k\\binom{3k}k}=81+16\\sqrt3\\,\\pi \\ \\ \\text{and}\\ \\ \\sum_{k=0}^\\infty\\frac{10k-1}{\\binom{4k}{2k}}=\\frac{4\\sqrt 3}{27}\\pi.$$\n  We also establish the following efficient formula for computing $\\log n$ with $1<n\\le 85/4$: \\begin{align*} &\\sum_{k=0}^\\infty\\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\\binom{4k}{2k}}\\\\ \\ \\ &=6n(n+1)(n-1)^3\\log n-32n(n+1)^2(n^2-4n+1), \\end{align*} where $$P(n):=n^6-58"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.08275","kind":"arxiv","version":7},"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/2204.08275/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":"2204.08275","created_at":"2026-07-05T10:22:41.361389+00:00"},{"alias_kind":"arxiv_version","alias_value":"2204.08275v7","created_at":"2026-07-05T10:22:41.361389+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.08275","created_at":"2026-07-05T10:22:41.361389+00:00"},{"alias_kind":"pith_short_12","alias_value":"UWBUZEDE2BFP","created_at":"2026-07-05T10:22:41.361389+00:00"},{"alias_kind":"pith_short_16","alias_value":"UWBUZEDE2BFPMXQC","created_at":"2026-07-05T10:22:41.361389+00:00"},{"alias_kind":"pith_short_8","alias_value":"UWBUZEDE","created_at":"2026-07-05T10:22:41.361389+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/UWBUZEDE2BFPMXQCBTH54M7LYR","json":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR.json","graph_json":"https://pith.science/api/pith-number/UWBUZEDE2BFPMXQCBTH54M7LYR/graph.json","events_json":"https://pith.science/api/pith-number/UWBUZEDE2BFPMXQCBTH54M7LYR/events.json","paper":"https://pith.science/paper/UWBUZEDE"},"agent_actions":{"view_html":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR","download_json":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR.json","view_paper":"https://pith.science/paper/UWBUZEDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2204.08275&json=true","fetch_graph":"https://pith.science/api/pith-number/UWBUZEDE2BFPMXQCBTH54M7LYR/graph.json","fetch_events":"https://pith.science/api/pith-number/UWBUZEDE2BFPMXQCBTH54M7LYR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR/action/storage_attestation","attest_author":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR/action/author_attestation","sign_citation":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR/action/citation_signature","submit_replication":"https://pith.science/pith/UWBUZEDE2BFPMXQCBTH54M7LYR/action/replication_record"}},"created_at":"2026-07-05T10:22:41.361389+00:00","updated_at":"2026-07-05T10:22:41.361389+00:00"}