Generalized binomial expansions and bernoulli polynomials

We investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. In particular, we present a new combinatorial formula for such sequences in terms of a ‘shift by rank’ quasi-expansion based on ordered set partitions. As an application, we give a new proof of Dilcher’s formula for expressing generalized Bernoulli polynomials in terms of classical Bernoulli polynomials.