Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles

Convolutions of Tribonacci, Fuss-Catalan, and Motzkin sequences

We introduce a class of sequences, defined by means of partial Bell polynomials, that contains a basis for the space of linear recurrence sequences with constant coefficients as well as other well-known sequences like Catalan and Motzkin. For the family of `Bell sequences' considered in this paper, we give a general multifold convolution formula and illustrate our result with a few explicit examples.

Catalan and Motzkin Integral Representations

We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in the literature are proved by using advanced mathematical techniques. All integral representations in this paper are proved by using standard techniques from integral calculus. Thus, we provide a more simplistic approach of proving integral representations of the Catalan and Motzkin numbers.

Exotic series with Bernoulli, harmonic, Catalan, and Stirling numbers

In this paper, we present a formula for generating various exotic series in the spirit of Ovidiu Furdui and Alina Sintamarian. Our new series (evaluated in closed form) involve Bernoulli, harmonic, and Catalan numbers. Also Stirling numbers of the second kind, other special numbers, and exponential polynomials. The results include series identities with Laguerre polynomials and derangement polynomials.

Motzkin 数及相关序列模 $p$ 的一致递归性

Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$

本文研究可表示为 Laurent 多项式常数项形式的 Motzkin 数等整数序列，刻画其模 $p$ 一致递归的素数，并将其余情形归约为性质良好的广义中心三项式系数。

Catalan Moments

This paper is essentially devoted to the study of some interesting relations among the well known operators $I^{(x)}$ (the interpolated Invert), $L^{(x)}$ (the interpolated Binomial) and Revert (that we call $η$). We prove that $I^{(x)}$ and $L^{(x)}$ are conjugated in the group $Υ(R)$. Here $R$ is a commutative unitary ring. In the same group we see that $η$ transforms $I^{(x)}$ in $L^{(-x)}$ by conjugation. These facts are proved as corollaries of much more general results. Then we carefully analyze the action of these operators on the set $\mc{R}$ of second order linear recurrent sequences.…

Congruence properties of combinatorial sequences via Walnut and the Rowland-Yassawi-Zeilberger automaton

Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which is an implementation of a decision procedure for proving various properties of automatic sequences. In this paper we explore some results (old and new) that can be proved using this method.