Sum of powers from uniform distribution moments

Abstract The sum of powers formulas for the first (n) consecutive integers may be viewed as the moments of discrete uniform distributions. These sum of powers equations are extended here for arbitrary moment configurations of the symmetric coefficients in a generalized Pascal's triangle. Moment equations are also derived for alternating signed versions of these coefficient moments. Moment generating functions for the sum of powers equations are presented in closed and in exponential sum formats. The moment equations and their exponential generating functions are expressed in terms of generalized Bernoulli, Euler and Chebyshev Polynomials, and by the Lerch transcendentals of the Hurwitz zeta and the Dirichlet eta functions. Following Faulhaber's lead, moment equations are written as power series of the term number (n). Finally, sum of powers tables are listed to facilitate moment calculations and to highlight coefficient characteristics in the moment equations.