Moments of Catalan Triangle Numbers

Abstract

In this chapter, we consider the Catalan numbers, C n ¼ 1 nþ1 2n n , and two of their generalizations, Catalan triangle numbers, B n,k and A n,k , for n, k ∈ . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: P n k¼1 k m B j n,k , P nþ1 k¼1 2k À 1 ðÞ m A j n,k , for j, n ∈  and m ∈  ∪ 0 fg. We present their closed expressions for some values of m and j. Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

Powers of Catalan generating functions for bounded operators

“…An alternative proof appears in Shapiro [4, Proposition 3.1]. Similarly for $$$ z=-1 $$$, we get that $$$$ \sum \limits_{k=1}^n{\left(-1\right)}^k{B}_{n,k}=-{C}_{n-1},\kern1em \sum \limits_{k=1}^{n+1}{\left(-1\right)}^k{A}_{n,k}=0,\kern1em n\ge 1, $$$$ see, for example, Miana and Romero [11, Theorems 2.1 and 2.2] and references therein. For $$$ z=\frac{1}{4} $$$, we follow similar ideas by induction method to get that $$$$ \sum \limits_{k=1}^n{B}_{n,k}{\left(\frac{1}{4}\right)}^k=\frac{a(n)}{4^n},\kern2em \sum \limits_{k=1}^{n+1}{A}_{n,k}{\left(\frac{1}{4}\right)}^k=\frac{b(n)}{4^{n+1}}, ...$$…”