We consider an Abel equation (*)y´ = p(x)y²+q(x)y³ with p(x), q(x) - polynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is that y₀ = y(0) y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients v_k(1) in the development . Following Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear) we introduce periods of the equation (*) as those , for which y(0) y() for any solution y(x) of (*). The generalized centre conditions are conditions on p, q under which given a₁, ... ,a_k are (exactly all) the periods of (*). A new basis for the ideals I_k = {v₂, ... ,v_k} has been produced in Briskin et al (1998 The Bautin ideal of the Abel equation Nonlinearity 10), defined by a linear recurrence relation. Using this basis and a special representation of polynomials, we extend results of Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), proving for small degrees of p and q a composition conjecture, as stated in Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129-52), Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), Briskin et al (Center Conditions II: Parametric and Model Centre Problems to appear). In particular, this provides transparent generalized centre conditions in the cases considered. We also compute maximal possible multiplicity of the zero solution of (*), extending the results of Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129-52).