{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:43:40Z","timestamp":1760111020199,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T00:00:00Z","timestamp":1718323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhoukou Normal University","award":["ZKNUC2022007","KYCX24_0725"],"award-info":[{"award-number":["ZKNUC2022007","KYCX24_0725"]}]},{"name":"Jiangsu Province","award":["ZKNUC2022007","KYCX24_0725"],"award-info":[{"award-number":["ZKNUC2022007","KYCX24_0725"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Making use of a telescoping approach, three types of sums of binomial quotients are examined. The summation terms of the two types of alternating sums have symmetry (i.e., their numerators and denominators are completely symmetric). We obtained a series of their explicit sums. Furthermore, by means of binomial relations, three recurrence relations of the sums are derived. In addition, series of double summation formulae involving binomial quotients are established.<\/jats:p>","DOI":"10.3390\/sym16060746","type":"journal-article","created":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T10:42:34Z","timestamp":1718361754000},"page":"746","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combinatorial Identities Concerning Binomial Quotients"],"prefix":"10.3390","volume":"16","author":[{"given":"Yulei","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9226-254X","authenticated-orcid":false,"given":"Dongwei","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gould, H.W. 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