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Among many others, we prove that \\begin{equation*}\n  \\sum_{k=1}^{n}\\frac{(-1)^{k-1}}{k}\\binom{n}{k}H_{n-k}=H_n^2+\\sum_{k=1}^{n}\\frac{(-1)^{k}}{k^2\\binom{n}{k}}, \\end{equation*} and \\begin{equation*} \\sum_{k=1}^{n}\\frac{(-1)^{k-1}}{k^2}\\binom{n}{k}H_{n-k}=\\frac{H_n[H_n^2+H_n^{(2)}]}{2}-\\sum_{k=0}^{n-1}\\frac{(-1)^k[H_n-H_k]}{(k+1)(n-k)\\binom{n}{k}}. \\end{equation*} Almost all of our results are new, while a few of them recapture know results."},"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":"1806.03022","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-08T08:37:59Z","cross_cats_sorted":[],"title_canon_sha256":"b31f26accc897e23476cd3a4618ca1736515106c4e6df6537d8da0892dffcf25","abstract_canon_sha256":"aa17a9ca922913aeb666104dba3a368d4c35e7ff6bc6384556e71c6db9ac465f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:50.185558Z","signature_b64":"Ssd1PDm82rBNfIHNr2/6ZYPiFCv2BBJmnVjfVZZgo+gqHRmWunO0SseTLouiNZB4F1IuEjiI7edp0CGHjQr7Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9db86cd75b7dae9918122588d23d5332d63185331aa6b103f13baad3bd380fca","last_reissued_at":"2026-05-18T00:13:50.184807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:50.184807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorial identities involving harmonic numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Necdet Batir","submitted_at":"2018-06-08T08:37:59Z","abstract_excerpt":"In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \\begin{equation*}\n  \\sum_{k=1}^{n}\\frac{(-1)^{k-1}}{k}\\binom{n}{k}H_{n-k}=H_n^2+\\sum_{k=1}^{n}\\frac{(-1)^{k}}{k^2\\binom{n}{k}}, \\end{equation*} and \\begin{equation*} \\sum_{k=1}^{n}\\frac{(-1)^{k-1}}{k^2}\\binom{n}{k}H_{n-k}=\\frac{H_n[H_n^2+H_n^{(2)}]}{2}-\\sum_{k=0}^{n-1}\\frac{(-1)^k[H_n-H_k]}{(k+1)(n-k)\\binom{n}{k}}. \\end{equation*} Almost all of our results are new, while a few of them recapture know results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03022","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":"1806.03022","created_at":"2026-05-18T00:13:50.184921+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.03022v1","created_at":"2026-05-18T00:13:50.184921+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.03022","created_at":"2026-05-18T00:13:50.184921+00:00"},{"alias_kind":"pith_short_12","alias_value":"TW4GZV23PWXJ","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"TW4GZV23PWXJSGAS","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"TW4GZV23","created_at":"2026-05-18T12:32:56.356000+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.19904","citing_title":"IBIS: Inverse BInomial sum Solver","ref_index":63,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL","json":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL.json","graph_json":"https://pith.science/api/pith-number/TW4GZV23PWXJSGASEWENEPKTGL/graph.json","events_json":"https://pith.science/api/pith-number/TW4GZV23PWXJSGASEWENEPKTGL/events.json","paper":"https://pith.science/paper/TW4GZV23"},"agent_actions":{"view_html":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL","download_json":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL.json","view_paper":"https://pith.science/paper/TW4GZV23","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.03022&json=true","fetch_graph":"https://pith.science/api/pith-number/TW4GZV23PWXJSGASEWENEPKTGL/graph.json","fetch_events":"https://pith.science/api/pith-number/TW4GZV23PWXJSGASEWENEPKTGL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL/action/storage_attestation","attest_author":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL/action/author_attestation","sign_citation":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL/action/citation_signature","submit_replication":"https://pith.science/pith/TW4GZV23PWXJSGASEWENEPKTGL/action/replication_record"}},"created_at":"2026-05-18T00:13:50.184921+00:00","updated_at":"2026-05-18T00:13:50.184921+00:00"}