Combinatorial proofs of some properties of tangent and Genocchi numbers. (June 2018) - Han, Guo-Niu; Liu, Jing-Yi
Abstract: The tangent number T 2 n + 1 is equal to the number of increasing labelled complete binary trees with 2 n + 1 vertices. This combinatorial interpretation immediately proves that T 2 n + 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 2 n + 1 by 2 2 n . 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.