A summation on Bernoulli numbers

In this paper, our aim is to investigate the summation form of Bernoulli numbers Bn, such as ∑k=0nnkBk+m. We derive some basic identities among them. These numbers can form a Seidel matrix. The upper diagonal elements of this Seidel matrix are called “the median Bernoulli numbers”. We determine the prime divisors of their numerators and denominators. And we characterize their ordinary generating function as the unique solution of some functional equation. At last, we also obtain the continued fraction representation of their ordinary generating function and their value of Hankel determinant.