On the Bernoulli Numbers via the Newton-Girard Identities

We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a sequence of linear operators on the space of functions. We prove properties of these polynomials, such as the positivity of their coefficients, and present a combinatorial formula for the Bernoulli numbers as a positive sum over plane trees which can be generalized as a transform of sequences. We also combinatorially prove the Newton-Girard identities using the symmetric group.