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arXiv · 1707.08882

Combinatorial proofs of some properties of tangent and Genocchi numbers

Guo-Niu Han·Jing-Yi Liu
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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.

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Guo-Niu Han, Jing-Yi Liu. 2017-07-27. Combinatorial proofs of some properties of tangent and Genocchi numbers. https://arxiv.org/abs/1707.08882

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