In this paper, for every $n \in \mathbb{ N}$, the following relationships between the functions $K_{ b}(n)$ and $K_{ e}(n)$ and the Bernoulli and Euler numbers are proved: \[ B_{ 2n} = -\,\frac{ (2n)!}{ 2^{ 2n}-2}\, K_{ b}(n), \qquad E_{ 2n} = (2n)!\, K_{ e}(n). \] The functions $K_{ b}$ and $K_{ e}$ are defined recursively by \[ K_{ b}(0) = K_{ e}(0) = 1, \] \[ K_{ b}(n) = - \sum_{ n'=0}^{ \,n-1} \frac{ K_{ b}(n')}{ \bigl( 2(n-n') + 1 \bigr)!}, \qquad n \ge 1, \] \[ K_{ e}(n) = - \sum_{ n'=0}^{ \,n-1} \frac{ K_{ e}(n')}{ \bigl( 2(n-n') \bigr)!}, \qquad n \ge 1. \] Furthermore, we present combinatorial interpretations of these functions in terms of ordered partitions of $n$: \[ K_{ b}(n) = \sum_{ \lambda \vDash n} \frac{ (-1)^{ \ell(\lambda)}} { \displaystyle\prod_{ i=1}^{ \ell(\lambda)} (2b_i + 1)!}, \qquad n \ge 1, \] \[ K_{ e}(n) = \sum_{ \lambda \vDash n} \frac{ (-1)^{ \ell(\lambda)}} { \displaystyle\prod_{ i=1}^{ \ell(\lambda)} (2b_i)!}, \qquad n \ge 1, \] where $\lambda = (b_1,b_2,\ldots,b_k) \vDash n$ and $\ell(\lambda)=k$.
Categories: math.GM