[Paper Review] Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles

José Luis Arregui|ArXiv.org|Sep 17, 2001

Advanced Combinatorial Mathematics7 references6 citations

TL;DR

This paper establishes a novel connection between Bernoulli and tangent numbers and combinatorial structures—Motzkin and Catalan numbers—via matrix-generated numerical triangles. By defining specific infinite matrices, the authors derive triangles where the first column yields tangent numbers and the diagonal entries relate to Bernoulli numbers through zeta function values, offering a new combinatorial interpretation of ζ(2n) using Motzkin paths and a matrix-based recurrence system.

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.

Motivation & Objective

• To establish a new combinatorial framework linking Bernoulli and tangent numbers to Motzkin and Catalan numbers.
• To demonstrate that tangent numbers arise as the first column of a matrix-generated numerical triangle.
• To show that ζ(2n) can be expressed using Motzkin paths and their enumeration via a specific matrix triangle.
• To unify the study of Bernoulli, tangent, and secant numbers through matrix-based numerical triangles.
• To extend Entringer’s work on alternating permutations by embedding it within a matrix-triangle formalism.

Proposed method

• Define a sequence of matrices A(n) with n rows and n+1 columns, generating a numerical triangle T = (tn,m) via matrix multiplication: tn = tn−1A(n).
• Use the matrix A with entries aij = j(j+1) if i ≥ j−1, and 0 otherwise, to generate a triangle whose first column gives tangent numbers tan(2n−1)(0).
• Express ζ(2n) combinatorially as a sum over Motzkin paths, using a volume interpretation inspired by Calabi’s method.
• Construct a matrix A(x) with variable x to generate a polynomial Pn(x) such that Pn(1) = (n+1)! and Pn(0) = tan(n+1)(0), enabling numerical extraction of tangent numbers.
• Apply the Entringer recurrence En+1,k+1 = En+1,k + En,n−k to generate the Seidel-Entringer-Arnold triangle via matrix submatrices.
• Use matrix products to recover both secant and tangent numbers as specific entries in the triangle generated by a modified matrix A.

Experimental results

Research questions

• RQ1 Can tangent and Bernoulli numbers be systematically generated via matrix-based numerical triangles?
• RQ2 How can ζ(2n) be expressed combinatorially using Motzkin paths and their enumeration?
• RQ3 What is the role of matrix structure in generating sequences like Catalan, Motzkin, and tangent numbers?
• RQ4 Can the Entringer recurrence be embedded in a matrix-triangle framework to recover both secant and tangent numbers?
• RQ5 Is there a polynomial interpolation method using matrix-generated triangles to extract tangent numbers from Pn(x) at x=0?

Key findings

• The first column of the triangle generated by the matrix A with aij = j(j+1) for i ≥ j−1 yields the sequence of tangent numbers tan(2n−1)(0).
• The diagonal entries of the same triangle are related to Bernoulli numbers via ζ(2n), with ζ(2n) expressible as a sum over Motzkin paths.
• A polynomial Pn(x) generated from a matrix A(x) satisfies Pn(1) = (n+1)! and Pn(0) = tan(n+1)(0), enabling numerical extraction of tangent numbers via small x values.
• The matrix A(x) with entries involving x and j(j+1) generates a triangle where Pn(x) approximates tan(n+1)(0) with error less than 1 when x = 1/(n+1)!.
• The Seidel-Entringer-Arnold triangle for alternating permutations can be reconstructed via matrix submatrices A(n) with entries 1 if i+j > n, 0 otherwise.
• The matrix A with entries 1 if i odd and i≤j, or j odd and j≤i+1, generates a triangle where each row is a permutation of Entringer numbers, and the first column gives zig-zag numbers βn.