Bernoulli numbers are especial case of Poly-Bernoulli numbers which are very important in combinatorics.
In definition of Poly-Bernoulli numbers
$${Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}$$
he used of poly-logarithm which has some relationships with poly-gamma , where The $B_{n}^{(1)}$ are the usual Bernoulli numbers.
A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of n by k (0,1)-matrices uniquely reconstructible from their row and column sums.
Here you can find combinatorics of Poly-Bernoulli numbers
For $\zeta(2n+1)$ Poly-Bernoulli numbers are more functional , see here