We introduce a new algebraic framework based on aromatic trees and Butcher-series for the systematic study of the accuracy of numerical integrators for sampling the invariant measure of a class of ergodic stochastic differential equations in R^d or on manifolds. In
particular, these new B-series, called exotic aromatic B-series, allow us to write conveniently Talay-Tubaro expansions, to perform
integrations by parts against the invariant measure, and satisfy geometric and algebraic properties.