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We extend this result by proving that, indeed, $$ \\sum_{i+j=n} \\binom{ai+k-\\ell}{i} \\binom{aj+\\ell}{j} = \\sum_{i+j=n} \\binom{ai+k}{i} \\binom{aj}{j} $$ for every integer $a$ and for every real $k$, and present new expressions for this value."},"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":"1302.2100","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-02-08T17:59:09Z","cross_cats_sorted":[],"title_canon_sha256":"718e487b104db238957484c3d20458f4413cdc8868480c6658089d0e7c8157a1","abstract_canon_sha256":"11bd38c2ed272330211caf65e35bd3ecc7f5832967f26532bcef4414d10f8c3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:38.658372Z","signature_b64":"Y9P0BWBfDjevxSN4XGgwWfigKpmmIEtHAVBVmrva6TEdd0ASEiqa1et+DW8IYrzMkjLxXI4l/i8m+DtTpXehCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98e757a360f08f97e5ac0736d2d4792f926d4d50bd168d4f5c0c053fb0d2f0c9","last_reissued_at":"2026-05-18T00:57:38.657630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:38.657630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Short note on the convolution of binomial coefficients","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":"2013-02-08T17:59:09Z","abstract_excerpt":"We know [Rui Duarte and Ant\\'onio Guedes de Oliveira, New developments of an old identity, manuscript arXiv:1203.5424, submitted.] that, for every non-negative integer numbers $n,i,j$ and for every real number $\\ell$, $$ \\sum_{i+j=n} \\binom{2i-\\ell}{i} \\binom{2j+\\ell}{j} = \\sum_{i+j=n}\\binom{2i}{i} \\binom{2j}{j}, $$ which is well-known to be $4^n$. We extend this result by proving that, indeed, $$ \\sum_{i+j=n} \\binom{ai+k-\\ell}{i} \\binom{aj+\\ell}{j} = \\sum_{i+j=n} \\binom{ai+k}{i} \\binom{aj}{j} $$ for every integer $a$ and for every real $k$, and present new expressions for this value."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2100","kind":"arxiv","version":1},"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":"1302.2100","created_at":"2026-05-18T00:57:38.657754+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.2100v1","created_at":"2026-05-18T00:57:38.657754+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.2100","created_at":"2026-05-18T00:57:38.657754+00:00"},{"alias_kind":"pith_short_12","alias_value":"TDTVPI3A6CHZ","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"TDTVPI3A6CHZPZNM","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"TDTVPI3A","created_at":"2026-05-18T12:28:02.375192+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/TDTVPI3A6CHZPZNMA43NFVDZF6","json":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6.json","graph_json":"https://pith.science/api/pith-number/TDTVPI3A6CHZPZNMA43NFVDZF6/graph.json","events_json":"https://pith.science/api/pith-number/TDTVPI3A6CHZPZNMA43NFVDZF6/events.json","paper":"https://pith.science/paper/TDTVPI3A"},"agent_actions":{"view_html":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6","download_json":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6.json","view_paper":"https://pith.science/paper/TDTVPI3A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.2100&json=true","fetch_graph":"https://pith.science/api/pith-number/TDTVPI3A6CHZPZNMA43NFVDZF6/graph.json","fetch_events":"https://pith.science/api/pith-number/TDTVPI3A6CHZPZNMA43NFVDZF6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6/action/storage_attestation","attest_author":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6/action/author_attestation","sign_citation":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6/action/citation_signature","submit_replication":"https://pith.science/pith/TDTVPI3A6CHZPZNMA43NFVDZF6/action/replication_record"}},"created_at":"2026-05-18T00:57:38.657754+00:00","updated_at":"2026-05-18T00:57:38.657754+00:00"}