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Sokal, Jes\\'us Salas","submitted_at":"2020-08-07T10:07:30Z","abstract_excerpt":"We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\\alpha n + \\beta k + \\gamma)\\, T(n-1,k)+(\\alpha' n + \\beta' k + \\gamma') \\, T(n-1,k-1)$ with initial condition $T(0,k) = \\delta_{k0}$ and parameters $\\mathbf{\\mu} = (\\alpha,\\beta,\\gamma, \\alpha',\\beta',\\gamma')$. We show that the family of arrays $T(\\mathbf{\\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \\times D_4$. Our main result is to determine all parameter sets $\\mathbf{\\mu} \\in \\mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \\sum_{n,k=0}^\\infty T(n,k) \\"},"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":"2008.03070","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-08-07T10:07:30Z","cross_cats_sorted":[],"title_canon_sha256":"c593ba59338180e3edb742f31bf898a5b19d08c7ec8c64e001392867c7df493c","abstract_canon_sha256":"b85f3056f24a2c5d42844432c8800362b0f8d18224db92cbeb600dfe403e004d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:38:29.832912Z","signature_b64":"WG+6Le9pqa4hJSqjTXfdZ4kWohxdvIQEexwKDoTV4ThWzC7QXV7jQhE2OU8O4XDwbr/WA6/qZ6H+9BSqS3GlDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7867c9b5a187c5f066d7285ee508fb589b6b6d15a0d390649349b973fb2a8fed","last_reissued_at":"2026-07-05T02:38:29.832540Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:38:29.832540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan D. Sokal, Jes\\'us Salas","submitted_at":"2020-08-07T10:07:30Z","abstract_excerpt":"We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\\alpha n + \\beta k + \\gamma)\\, T(n-1,k)+(\\alpha' n + \\beta' k + \\gamma') \\, T(n-1,k-1)$ with initial condition $T(0,k) = \\delta_{k0}$ and parameters $\\mathbf{\\mu} = (\\alpha,\\beta,\\gamma, \\alpha',\\beta',\\gamma')$. We show that the family of arrays $T(\\mathbf{\\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \\times D_4$. Our main result is to determine all parameter sets $\\mathbf{\\mu} \\in \\mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \\sum_{n,k=0}^\\infty T(n,k) \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.03070","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.03070/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":"2008.03070","created_at":"2026-07-05T02:38:29.832598+00:00"},{"alias_kind":"arxiv_version","alias_value":"2008.03070v2","created_at":"2026-07-05T02:38:29.832598+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.03070","created_at":"2026-07-05T02:38:29.832598+00:00"},{"alias_kind":"pith_short_12","alias_value":"PBT4TNNBQ7C7","created_at":"2026-07-05T02:38:29.832598+00:00"},{"alias_kind":"pith_short_16","alias_value":"PBT4TNNBQ7C7AZWX","created_at":"2026-07-05T02:38:29.832598+00:00"},{"alias_kind":"pith_short_8","alias_value":"PBT4TNNB","created_at":"2026-07-05T02:38:29.832598+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/PBT4TNNBQ7C7AZWXFBPOKCH3LC","json":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC.json","graph_json":"https://pith.science/api/pith-number/PBT4TNNBQ7C7AZWXFBPOKCH3LC/graph.json","events_json":"https://pith.science/api/pith-number/PBT4TNNBQ7C7AZWXFBPOKCH3LC/events.json","paper":"https://pith.science/paper/PBT4TNNB"},"agent_actions":{"view_html":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC","download_json":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC.json","view_paper":"https://pith.science/paper/PBT4TNNB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2008.03070&json=true","fetch_graph":"https://pith.science/api/pith-number/PBT4TNNBQ7C7AZWXFBPOKCH3LC/graph.json","fetch_events":"https://pith.science/api/pith-number/PBT4TNNBQ7C7AZWXFBPOKCH3LC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC/action/storage_attestation","attest_author":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC/action/author_attestation","sign_citation":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC/action/citation_signature","submit_replication":"https://pith.science/pith/PBT4TNNBQ7C7AZWXFBPOKCH3LC/action/replication_record"}},"created_at":"2026-07-05T02:38:29.832598+00:00","updated_at":"2026-07-05T02:38:29.832598+00:00"}