Best constants for a family of Carleson sequences

We consider a general family of Carleson sequences associated with dyadic A2A2 weights and find sharp — or, in one case, simply best known — upper and lower bounds for their Carleson norms in terms of the A2A2-characteristic of the weight. The results obtained make precise and significantly generalize earlier estimates by Wittwer, Vasyunin, Beznosova, and others. We also record several corollaries, one of which is a range of new characterizations of dyadic A2A2. Particular emphasis is placed on the relationship between sharp constants and optimizing sequences of weights; in most cases explicit optimizers are constructed. Our main estimates arise as consequences of the exact expressions, or explicit bounds, for the Bellman functions for the problem, and the paper contains a measure of Bellman-function innovation.