  We show that each member of a doubly infinite sequence of highly nonlinear
expressions of Bernoulli polynomials, which can be seen as linear combinations
of certain higher-order convolutions, is a multiple of a specific product of
linear factors. The special case of Bernoulli numbers has important
applications in the study of multiple Tornheim zeta functions. The proof of the
main result relies on properties of Eulerian polynomials and higher-order
Bernoulli polynomials.
