Sums of powers of Catalan triangle numbers

  1. Miana, P.J. 1
  2. Ohtsuka, H. 2
  3. Romero, N. 3
  1. 1 Universidad de Zaragoza
    info

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

  2. 2 Bunkyo University High School, 1191-7, Kami, Ageo-city, Saitama Pref., Japan
  3. 3 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

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Revue:
Discrete Mathematics

ISSN: 0012-365X

Année de publication: 2017

Volumen: 340

Número: 10

Pages: 2388-2397

Type: Article

DOI: 10.1016/J.DISC.2017.05.006 SCOPUS: 2-s2.0-85020549149 WoS: WOS:000407182200007 GOOGLE SCHOLAR

D'autres publications dans: Discrete Mathematics

Projets liés

Ecuaciones no lineales y métodos iterativos. Aplicaciones a problemas de optimización y ecuaciones matriciales. Nonlinear equations and iterative methods. Applications to optimization problems and matrix equations.

2015/00069/001

Résumé

In this paper, we consider combinatorial numbers (Cm,k)m≥1,k≥0, mentioned as Catalan triangle numbers where Cm,k≔−1k−m−1k−1. These numbers unify the entries of the Catalan triangles Bn,k and An,k for appropriate values of parameters m and k, i.e., Bn,k=C2n,n−k and An,k=C2n+1,n+1−k. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers Cn that is C2n,n−1=C2n+1,n=Cn. We present identities for sums (and alternating sums) of Cm,k, squares and cubes of Cm,k and, consequently, for Bn,k and An,k. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between (Cm,k)m≥1,k≥0 and harmonic numbers (Hn)n≥1. Finally, in the last section, new open problems and identities involving (Cn)n≥0 are conjectured. © 2017 Elsevier B.V.