  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.
