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

TL;DR

This paper demonstrates how Bernoulli and tangent numbers can be derived from numerical triangles related to Catalan and Motzkin numbers, using a new expression of Zeta(2n) involving Motzkin paths.

Contribution

It introduces novel triangles of numbers that connect Bernoulli and tangent numbers with Catalan and Motzkin numbers through new combinatorial expressions.

Findings

Bernoulli and tangent numbers can be obtained from specific numerical triangles.

A new expression of Zeta(2n) involving Motzkin paths is presented.

Connections between classical number sequences and combinatorial triangles are established.

Abstract

It is shown that Bernoulli numbers and tangent numbers (the derivatives of the tangent function at zero) can be obtained by means of easily defined triangles of numbers in several ways, some of them very similar to the Catalan triangle and a Motzkin-like triangle. Our starting point in order to show this is a new expression of Zeta(2n) involving Motzkin paths.