Catalan's intriguing factorial problem

• Autores: Thomas Koshy
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 43, Nº. 1, 2012, págs. 101-109
• Idioma: inglés
• DOI: 10.1080/0020739x.2011.562322
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• Resumen

  • This article investigates the numbers , originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers Cn, we develop some divisibility properties for Cn. In particular, we establish that , where fkdenotes the kth Fermat number and Mkthe kth Mersenne number 2k?-?1. Finally, we develop an explicit formula for Xm,nusing Pascal's triangle and Catalan numbers, and extract several interesting byproducts from it.