{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:WCVPWX6HEPWAJDTKLBWWPFL2HM","short_pith_number":"pith:WCVPWX6H","canonical_record":{"source":{"id":"1008.3887","kind":"arxiv","version":13},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-23T19:07:36Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eb5a02292c550745a5f6cf588ea6e8dcd5f1b1e2247ed370d84fab88293c4729","abstract_canon_sha256":"3fcbf1c81a96f76ceabbc057473ff56e2bf484d71af32239c7d418d15c837796"},"schema_version":"1.0"},"canonical_sha256":"b0aafb5fc723ec048e6a586d67957a3b15daeb527424e2017307bd42a552943d","source":{"kind":"arxiv","id":"1008.3887","version":13},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3887","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3887v13","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3887","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"pith_short_12","alias_value":"WCVPWX6HEPWA","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WCVPWX6HEPWAJDTK","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WCVPWX6H","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:WCVPWX6HEPWAJDTKLBWWPFL2HM","target":"record","payload":{"canonical_record":{"source":{"id":"1008.3887","kind":"arxiv","version":13},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-23T19:07:36Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eb5a02292c550745a5f6cf588ea6e8dcd5f1b1e2247ed370d84fab88293c4729","abstract_canon_sha256":"3fcbf1c81a96f76ceabbc057473ff56e2bf484d71af32239c7d418d15c837796"},"schema_version":"1.0"},"canonical_sha256":"b0aafb5fc723ec048e6a586d67957a3b15daeb527424e2017307bd42a552943d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:39.189170Z","signature_b64":"iF6NY9ARIRo3ZVnGe/qwWY98L7O9Gy/3GAnvFgwaPGYIvdqsoYXdqt8AJ6lAERZg0koZrWIJPUHxrgEg7lb4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0aafb5fc723ec048e6a586d67957a3b15daeb527424e2017307bd42a552943d","last_reissued_at":"2026-05-18T02:49:39.188847Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:39.188847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.3887","source_version":13,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kyNhxMtOg1uhxyRXXIBCVeGjqrLBzYpdLElROlRiAWF4UGdW5yVGfRx8/kFaw9ezGjMtMt8ZawALumG5gjKICg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T23:06:45.288231Z"},"content_sha256":"051ce6b46b968a2f094e6f6fff9b348c13a86a93aee5571f5c9c15ff12c9b8f5","schema_version":"1.0","event_id":"sha256:051ce6b46b968a2f094e6f6fff9b348c13a86a93aee5571f5c9c15ff12c9b8f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:WCVPWX6HEPWAJDTKLBWWPFL2HM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Congruences involving generalized central trinomial coefficients","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":"2010-08-23T19:07:36Z","abstract_excerpt":"For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\\ (n=0,1,2,\\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\\ldots$ we have $$\\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\\equiv0\\pmod{n^2}$$ and in particular $n^2\\mi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3887","kind":"arxiv","version":13},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ml2nNfm+1PoZ7wgUBXL3dBV/xG9I8fGGZgZ9yokNVSB40lGVfF+Ptv8R7rA3CpgMoG7b9fKaiXvXgOcgt9ugCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T23:06:45.289045Z"},"content_sha256":"2908f067aaf59ab30ab3edb99d6812e18cdaaa49885945f4e9f765b993f4ddf3","schema_version":"1.0","event_id":"sha256:2908f067aaf59ab30ab3edb99d6812e18cdaaa49885945f4e9f765b993f4ddf3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM/bundle.json","state_url":"https://pith.science/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-22T23:06:45Z","links":{"resolver":"https://pith.science/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM","bundle":"https://pith.science/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM/bundle.json","state":"https://pith.science/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WCVPWX6HEPWAJDTKLBWWPFL2HM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:WCVPWX6HEPWAJDTKLBWWPFL2HM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3fcbf1c81a96f76ceabbc057473ff56e2bf484d71af32239c7d418d15c837796","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-23T19:07:36Z","title_canon_sha256":"eb5a02292c550745a5f6cf588ea6e8dcd5f1b1e2247ed370d84fab88293c4729"},"schema_version":"1.0","source":{"id":"1008.3887","kind":"arxiv","version":13}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3887","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3887v13","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3887","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"pith_short_12","alias_value":"WCVPWX6HEPWA","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WCVPWX6HEPWAJDTK","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WCVPWX6H","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:2908f067aaf59ab30ab3edb99d6812e18cdaaa49885945f4e9f765b993f4ddf3","target":"graph","created_at":"2026-05-18T02:49:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\\ (n=0,1,2,\\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\\ldots$ we have $$\\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\\equiv0\\pmod{n^2}$$ and in particular $n^2\\mi","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-23T19:07:36Z","title":"Congruences involving generalized central trinomial coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3887","kind":"arxiv","version":13},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:051ce6b46b968a2f094e6f6fff9b348c13a86a93aee5571f5c9c15ff12c9b8f5","target":"record","created_at":"2026-05-18T02:49:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3fcbf1c81a96f76ceabbc057473ff56e2bf484d71af32239c7d418d15c837796","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-23T19:07:36Z","title_canon_sha256":"eb5a02292c550745a5f6cf588ea6e8dcd5f1b1e2247ed370d84fab88293c4729"},"schema_version":"1.0","source":{"id":"1008.3887","kind":"arxiv","version":13}},"canonical_sha256":"b0aafb5fc723ec048e6a586d67957a3b15daeb527424e2017307bd42a552943d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0aafb5fc723ec048e6a586d67957a3b15daeb527424e2017307bd42a552943d","first_computed_at":"2026-05-18T02:49:39.188847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:39.188847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iF6NY9ARIRo3ZVnGe/qwWY98L7O9Gy/3GAnvFgwaPGYIvdqsoYXdqt8AJ6lAERZg0koZrWIJPUHxrgEg7lb4Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:39.189170Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.3887","source_kind":"arxiv","source_version":13}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:051ce6b46b968a2f094e6f6fff9b348c13a86a93aee5571f5c9c15ff12c9b8f5","sha256:2908f067aaf59ab30ab3edb99d6812e18cdaaa49885945f4e9f765b993f4ddf3"],"state_sha256":"95f6f1bf065c2fed119fb9e7e56fb73b581bcb06f1b9a7040f3068b1f8e6645c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oPw/T1NLVO4uG4y7bauxyHrLMBLGkLXl4HR7T7Nhd1HB5H8NmvxVSG8J+pDHn++6f2vPNgwXImoQGEBvaG0xAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T23:06:45.294304Z","bundle_sha256":"1cab246d7aa4c694a2579975ccbeccba8d92fd73b4b9f71dd96874826eb36991"}}