Moments of combinatorial and Catalan numbers
type: Item
instance of: scholarly article
title: Moments of combinatorial and Catalan numbers
zbMATH Open document ID: 1214.11033
author: Natalia Romero
author: Pedro J. Miana
published in: Journal of Number Theory
publication date: 2010-07-07T00:00:00Z
review text: If \(n\) is a natural number, then the \(n\)th Catalan number is defined by \[ C_n= {1\over n+1} {3n\choose n}. \] If also the natural number \(p\geq n\), define \[ B_{n,p}= {p\over n} {2n\choose n-p};\quad A_{n,p}= {2p-1\over 2n+1} {2n+1\choose n+1-p}. \] Let \(\Omega_m(n)= \sum^n_{p=1} p^m B^2_{n,p}\). The authors obtain formulas that express \(\Omega_m(n)\) in terms of Catalan numbers and polynomial sequences with integer coefficients. They also obtain formulas for the additional functions listed below: \[ \Phi_m(n)= \sum^{n+1}_{p=1} (2p- 1)^m{2n+1\choose n+1-p}^2;\quad \Psi_m(n)= \sum^{n+1}_{p=1} (2p-1)^m A^2_{n,p}. \] (There are actually separate formulas corresponding to the cases \(m\) odd, \(m\) even.)
reviewed by: Neville Robbins
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 5731640
zbMATH Keywords: Catalan numbers
zbMATH Keywords: combinatorial identities
zbMATH Keywords: binomial coefficients
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.01.018
OpenAlex ID: W1993530625
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DOI: 10.1016/J.JNT.2010.01.018
description: scientific article; zbMATH DE number 5731640