Shifted sums of the Bernoulli numbers, reciprocity, and denominators

We consider the numbers ${\mathcal{B}}_{r,s}=(\mathbf{B}+1{)}^r{\mathbf{B}}^s$ (in umbral notation ${\mathbf{B}}^n={\mathbf{B}}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the literature and goes back to the 19th century. In a recent paper, self-reciprocal Bernoulli polynomials, whose coefficients are related to these numbers, appeared in the context of power sums and the so-called Faulhaber polynomials. The numbers ${\mathcal{B}}_{r,s}$ can be recursively expressed by iterated sums and differences, so it is not obvious that these numbers do not vanish in general. As a main result among other properties, we show the non-vanishing of these numbers, apart from exceptional cases. We further derive an explicit product formula for their denominators, which follows from a von Staudt–Clausen type relation.