Moments on Catalan numbers

Abstract

By combining inverse series relations with binomial convolutions and telescoping method, moments of Catalan numbers are evaluated, which resolves a problem recently proposed by Gutiérrez et al. [J.M. Gutiérrez, M.A. Hernández, P.

Factors of binomial sums from the Catalan triangle

“…Recently, Gutiérrez et al [9], Miana and Romero [11], and Chen and Chu [5] studied the binomial sums n k=1 k m 2n n−k 2 . On the other hand, Miana and Romero [11] have proved the following identity: where (a) n = a(a + 1) · · · (a + n − 1) is the Pochhammer symbol.…”

Moments of combinatorial and Catalan numbers

“…Now, from Lemma 1, there exists a polynomial of integer coefficients Q k,2m+3 and degree at most 2m + 3 such that 2 2m−k+3 λ k (m + 1, n + 1 2 )n(n − 1) · · · (n − k + 1) (2n + 1)(2n − 3)(2n − 5) · · · (2n + 1 − 2k)(2n + 1 − 2k) = 2 2m−2k+3 Q k,2m+3 (n) 2n + 1 − 2k for 2 k m + 1. Then, we conclude that Ψ 2m+1 (n) = (n + 1)C n C n−1 P 3m+2 (n) (2n − 3) · · · (2n − 2m − 1) , n ∈ N,…”

“…for m 2 and y ∈ R, see more details in [1,2]. In the following lemma we give interesting properties of these polynomials λ k (m, y).…”

“…In the first section of this paper, we improve some results presented in [1] to apply them to some questions posed in [6]. In the second and third sections, we consider the moments {Φ m } m 0 defined …”

Combinatorial identities related to 2 × 2 submatrices of recursive matrices

“…Shapiro's Catalan triangle, defined by B = (B n,k ) n≥k≥0 with B n,k = k+1 n+1 2n+2 n−k [20, A039598], appears in various combinatorial settings [2,17,18,19] and has been paid a lot attention in combinatorics and number theory [3,8,11,12,14,15,23,24,25]. Define another infinite lower triangle X = (X n,k ) n≥k≥0 , based on the triangle B as follows, X n,k = det B n,k B n,k+1 B n+1,k B n+1,k+1 .…”