We investigate various bases for the flag f-vectors of Eulerian posets.
Many of the change-of-basis formulas are seen to be triangular.
One change-of-basis formula implies the following:
If the Charney-Davis Conjecture is true for order complexes, then certain sums of cd-coefficients are non-negative in all Gorenstein* posets.
In particular, cd-coefficients with no adjacent c's are non-negative.
A convolution formula for cd-coefficients, together with the proof by M.~Davis and B.~Okun of the Charney-Davis Conjecture in dimension 3,
imply that certain additional cd-coefficients are non-negative for all Gorenstein* posets.
In particular we verify, up to rank 6, Stanley's conjecture that the coefficients in the cd-index of a Gorenstein* ranked poset are non-negative.