Skein-theoretical derivation of some formulas of Habiro

We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of sl_2 invariants of integral homology 3-spheres. Habiro originally derived these formulas using the quantum group U_q sl_2. As an application, we give a formula for the colored Jones polynomial of twist knots, generalizing formulas of Habiro and Le for the trefoil and the figure eight knot.