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We also consider the convolution identities with binomial coefficients: $$ \\sum_{k_1+\\cdots+k_r=n\\atop k_1,\\dots,k_r\\ge 1}\\binom{n}{k_1,\\dots,k_r}B_{k_1}\\cdots B_{k_r} $$ This type can be generalized, so that $B_n$ is a special case of the number $u_n$, where $u_n=a u_{n-1}+b u_{n-2}$ ($n\\ge 2$) with $u_0=0$ and $"},"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":"1608.05925","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-21T12:15:46Z","cross_cats_sorted":[],"title_canon_sha256":"cf5731b38fe004c7540687548743b983cf4467a77b3bcb57246a55903ce92627","abstract_canon_sha256":"78d07b64b395ea5fc4a0829d387247469a2aef30a8a4a7fa43669aac6aabc163"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:22.828522Z","signature_b64":"sOmVH8d0OH7ANEz3ft7IvjcQWklGozVBtXIASfa4Y7npg37LKwf/AEfhwK9Y7QLb54QLBKrAq6qIW8Ce5SFIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd64a7526b7101381d2a34c6e347f6c5955dccff3748d42b0e028cd9b7e11709","last_reissued_at":"2026-05-18T01:08:22.827945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:22.827945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher-order identities for balancing numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Prasanta Kumar Ray, Takao Komatsu","submitted_at":"2016-08-21T12:15:46Z","abstract_excerpt":"Let $B_n$ be the $n$-th balancing number. In this paper, we give some explicit expressions of $\\sum_{l=0}^{2 r-3}(-1)^l\\binom{2 r-3}{l}\\sum_{j_1+\\cdots+j_r=n-2 l\\atop j_1,\\dots,j_r\\ge 1}B_{j_1}\\cdots B_{j_r}$ and $\\sum_{j_1+\\cdots+j_r=n\\atop j_1,\\dots,j_r\\ge 1}B_{j_1}\\cdots B_{j_r}$. We also consider the convolution identities with binomial coefficients: $$ \\sum_{k_1+\\cdots+k_r=n\\atop k_1,\\dots,k_r\\ge 1}\\binom{n}{k_1,\\dots,k_r}B_{k_1}\\cdots B_{k_r} $$ This type can be generalized, so that $B_n$ is a special case of the number $u_n$, where $u_n=a u_{n-1}+b u_{n-2}$ ($n\\ge 2$) with $u_0=0$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05925","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":"1608.05925","created_at":"2026-05-18T01:08:22.828025+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.05925v1","created_at":"2026-05-18T01:08:22.828025+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.05925","created_at":"2026-05-18T01:08:22.828025+00:00"},{"alias_kind":"pith_short_12","alias_value":"XVSKOUTLOEAT","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"XVSKOUTLOEATQHJK","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"XVSKOUTL","created_at":"2026-05-18T12:30:51.357362+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/XVSKOUTLOEATQHJKGTDOGR7WYW","json":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW.json","graph_json":"https://pith.science/api/pith-number/XVSKOUTLOEATQHJKGTDOGR7WYW/graph.json","events_json":"https://pith.science/api/pith-number/XVSKOUTLOEATQHJKGTDOGR7WYW/events.json","paper":"https://pith.science/paper/XVSKOUTL"},"agent_actions":{"view_html":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW","download_json":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW.json","view_paper":"https://pith.science/paper/XVSKOUTL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.05925&json=true","fetch_graph":"https://pith.science/api/pith-number/XVSKOUTLOEATQHJKGTDOGR7WYW/graph.json","fetch_events":"https://pith.science/api/pith-number/XVSKOUTLOEATQHJKGTDOGR7WYW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW/action/storage_attestation","attest_author":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW/action/author_attestation","sign_citation":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW/action/citation_signature","submit_replication":"https://pith.science/pith/XVSKOUTLOEATQHJKGTDOGR7WYW/action/replication_record"}},"created_at":"2026-05-18T01:08:22.828025+00:00","updated_at":"2026-05-18T01:08:22.828025+00:00"}