Some combinatorial identities containing central binomial coefficients or Catalan numbers*

Abstract

In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers; establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers. The instant proof simplifies discussions in the journal papers: College Math. J. 43 (2012), no. 2, 141–146; Amer. Math. Monthly 121 (2014), no. 3, 267–267; Amer. Math. Monthly 123 (2016), no. 4, 405–406; Elem. Math. 71 (2016), no. 3, 109–121; and Mathematics 5 (2017), no. 3, Article 40, 31 pages.

Keywords

• catalan number
• arcsine
• square
• cubic
• ratio
• product
• sum formula
• central binomial coefficient
• finite sum
• maclaurin's expansion