Sums of powers of Catalan triangle numbers
| Autor: | Hideyuki Ohtsuka, Pedro J. Miana, Natalia Romero |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
05A19 05A10 11B65 Sums of powers Mathematics - Number Theory 010102 general mathematics 0102 computer and information sciences 01 natural sciences language.human_language Theoretical Computer Science Combinatorics Catalan number Section (category theory) Schröder–Hipparchus number 010201 computation theory & mathematics language FOS: Mathematics Discrete Mathematics and Combinatorics Harmonic number Catalan Number Theory (math.NT) 0101 mathematics Binomial coefficient Mathematics |
| DOI: | 10.48550/arxiv.1602.04347 |
| Popis: | In this paper, we consider combinatorial numbers ( C m , k ) m ≥ 1 , k ≥ 0 , mentioned as Catalan triangle numbers where C m , k ≔ m − 1 k − m − 1 k − 1 . These numbers unify the entries of the Catalan triangles B n , k and A n , k for appropriate values of parameters m and k , i.e., B n , k = C 2 n , n − k and A n , k = C 2 n + 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 C n that is C 2 n , n − 1 = C 2 n + 1 , n = C n . We present identities for sums (and alternating sums) of C m , k , squares and cubes of C m , k and, consequently, for B n , k and A n , k . In particular, one of these identities solves an open problem posed in Gutierrez et al. (2008). We also give some identities between ( C m , k ) m ≥ 1 , k ≥ 0 and harmonic numbers ( H n ) n ≥ 1 . Finally, in the last section, new open problems and identities involving ( C n ) n ≥ 0 are conjectured. |
| Databáze: | OpenAIRE |
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