{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:KYYK3X76XMVT2C42QJUJ5UIZFV","short_pith_number":"pith:KYYK3X76","schema_version":"1.0","canonical_sha256":"5630addffebb2b3d0b9a82689ed1192d59456ddab03a35e0e35d2b8d2ddb31ee","source":{"kind":"arxiv","id":"1203.5424","version":4},"attestation_state":"computed","paper":{"title":"New developments of an old identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Guedes de Oliveira, Rui Duarte","submitted_at":"2012-03-24T15:38:01Z","abstract_excerpt":"We give a direct combinatorial proof of a famous identity, $$ \\sum_{i+j=n} m{2i}{i} \\binom{2j}{j} = 4^n $$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss two different generalizations of the identity, and end the paper by presenting in explicit form the ordinary generating function of the sequence $(\\strut\\binom{2n+k}{n})_{n\\in\\mathds{N}_0}$, where $k\\in\\mathds{R}$."},"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":"1203.5424","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-24T15:38:01Z","cross_cats_sorted":[],"title_canon_sha256":"5544d4235f95dc077bb1c106c21ad24bd6c960b0bfa0c9e4a0acba596fce9d59","abstract_canon_sha256":"f17b1bc7048b3c723709719c3b9ccf3f535f4577d15b5ced2cc604c22ae68970"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:38.684939Z","signature_b64":"3BxvWoxW+xVa8PakD8UGSx4PkTjeUVgU3pTQ/7WEPmptQx8HguFpo4pFBMUzdPeXCnzF2iR3IPysTpAcdOH1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5630addffebb2b3d0b9a82689ed1192d59456ddab03a35e0e35d2b8d2ddb31ee","last_reissued_at":"2026-05-18T00:57:38.684098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:38.684098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New developments of an old identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Guedes de Oliveira, Rui Duarte","submitted_at":"2012-03-24T15:38:01Z","abstract_excerpt":"We give a direct combinatorial proof of a famous identity, $$ \\sum_{i+j=n} m{2i}{i} \\binom{2j}{j} = 4^n $$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss two different generalizations of the identity, and end the paper by presenting in explicit form the ordinary generating function of the sequence $(\\strut\\binom{2n+k}{n})_{n\\in\\mathds{N}_0}$, where $k\\in\\mathds{R}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5424","kind":"arxiv","version":4},"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":"1203.5424","created_at":"2026-05-18T00:57:38.684239+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.5424v4","created_at":"2026-05-18T00:57:38.684239+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5424","created_at":"2026-05-18T00:57:38.684239+00:00"},{"alias_kind":"pith_short_12","alias_value":"KYYK3X76XMVT","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"KYYK3X76XMVT2C42","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"KYYK3X76","created_at":"2026-05-18T12:27:11.947152+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/KYYK3X76XMVT2C42QJUJ5UIZFV","json":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV.json","graph_json":"https://pith.science/api/pith-number/KYYK3X76XMVT2C42QJUJ5UIZFV/graph.json","events_json":"https://pith.science/api/pith-number/KYYK3X76XMVT2C42QJUJ5UIZFV/events.json","paper":"https://pith.science/paper/KYYK3X76"},"agent_actions":{"view_html":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV","download_json":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV.json","view_paper":"https://pith.science/paper/KYYK3X76","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.5424&json=true","fetch_graph":"https://pith.science/api/pith-number/KYYK3X76XMVT2C42QJUJ5UIZFV/graph.json","fetch_events":"https://pith.science/api/pith-number/KYYK3X76XMVT2C42QJUJ5UIZFV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV/action/storage_attestation","attest_author":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV/action/author_attestation","sign_citation":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV/action/citation_signature","submit_replication":"https://pith.science/pith/KYYK3X76XMVT2C42QJUJ5UIZFV/action/replication_record"}},"created_at":"2026-05-18T00:57:38.684239+00:00","updated_at":"2026-05-18T00:57:38.684239+00:00"}