The Triangle Algorithm for Bernoulli Polynomials

Algorithms like ones used to generate Pascal's triangle for generating Bernoulli polynomials and their various generalizations are given.  It is remarkable that the algorithms for Bernoulli polynomials are natural interpolations of the ones for Bernoulli numbers. 
The algorithms presented in this paper can be understood as essentially unique, even if the Bernoulli polynomials are generalized in various ways.