{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/two-integer-sequences-related-to-catalan","title":"Two integer sequences related to Catalan numbers","arxiv_id":"1009.4225","date":"2010-09-21","proceeding":null,"authors":["Michel Lassalle"],"abstract":"We prove the following conjecture of Zeilberger. Denoting by $C_n$ the Catalan number, define inductively $A_n$ by $(-1)^{n-1}A_n=C_n+\\sum_{j=1}^{n-1} (-1)^{j} \\binom{2n-1}{2j-1} A_j \\,C_{n-j}$ and $a_n=2A_n/C_n$. Then $a_n$ (hence $A_n$) is a positive integer.","url_abs":"http://arxiv.org/abs/1009.4225v4","url_pdf":"http://arxiv.org/pdf/1009.4225v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"two-integer-sequences-related-to-catalan","repo_url":"https://github.com/JuliaMath/Combinatorics.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"two-integer-sequences-related-to-catalan","repo_url":"https://github.com/UnofficialJuliaMirror/Catalan.jl-a10504cf-72a7-53fd-8489-d9cc60862a4b","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"two-integer-sequences-related-to-catalan","repo_url":"https://github.com/UnofficialJuliaMirror/Combinatorics.jl-861a8166-3701-5b0c-9a16-15d98fcdc6aa","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"two-integer-sequences-related-to-catalan","repo_url":"https://github.com/UnofficialJuliaMirrorSnapshots/Combinatorics.jl-861a8166-3701-5b0c-9a16-15d98fcdc6aa","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}