Combinatorial proofs of some properties of tangent and Genocchi numbers

TL;DR

This paper offers a combinatorial proof for a divisibility property of tangent and Genocchi numbers using hook length formulas, and extends the approach to k-ary trees for new generalizations.

Contribution

The paper introduces a combinatorial proof of a known divisibility property and generalizes Genocchi numbers to k-ary trees.

Findings

Proved divisibility of (n+1)T_{2n+1} by 2^{2n} using combinatorial methods.

Extended the combinatorial approach to k-ary trees, leading to new Genocchi number generalizations.

Provided a more straightforward proof compared to traditional calculation-heavy methods.

Abstract

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers.