The tangent number $T_{2n+1}$ is equal to the number of increasing labeledcomplete binary trees with $2n+1$ vertices. This combinatorial interpretationimmediately proves that $T_{2n+1}$ is divisible by $2^n$. However a strongerdivisibility property is known in the studies of Bernoulli and Genocchinumbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. Thetraditional proofs of this fact need long calculations. In the present paper,we provide a combinatorial proof of the later divisibility by using the hooklength formula for trees. Furthermore, our method is extented to $k$-ary trees,which leads to a new generalization of the Genocchi numbers.
展开▼
