Two integer sequences related to Catalan numbers

Abstract

International audienceWe prove the following conjecture of Zeilberger. Denoting by C-n the Catalan number, define inductively A(n) by (-1)(n-1) A(n) = Cn + Sigma(n-1)(j=1) (-1)(j) ((2n-1)(2j-1))A(j)C(n-j) and a(n) = 2A(n)/C-n. Then a,, (hence A(n)) is a positive integer. (C) 2012 Elsevier Inc. All rights reserved

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

“…Let a n = 2A n /C n . Lasalle [12] and Amdeberhan et al [1] showed that both {A n } n≥1 and {a n } n≥2 are increasing sequences of positive integers. The latter also obtained the recurrence 2na n = n−1 k=1 n k − 1 n k + 1 a k a n−k , n = 2, 3, .…”

Recursion rules for the hypergeometric zeta function

“…for the class of functions u for which the integral is finite. The techniques employed here were recently used in [3] to solve a problem proposed by D. Zeilberger related to the Narayana polynomials discussed in [18]. Definition 6.1.…”

Proof of a positivity conjecture on Schur functions

“…The objective of this paper is to prove a positivity conjecture on Schur functions, which was proposed by Lassalle [5] in the study of two combinatorial sequences related to the Catalan numbers.…”

“…Let C n = 1 n + 1 2n n denote the n-th Catalan number. Lassalle [5] introduced a sequence of numbers A n for n ≥ 1, which are recursively defined by…”

“…By using the theory of symmetric functions, Lassalle [5] gave a direct proof of the positivity and the monotonicity of {A n } n≥2 , although these two properties can be deduced from Zeilberger's conjecture. For the notation and terminology on symmetric functions, see Stanley [11].…”