  We present a recursive method to calculate a large q expansion of the 2d
q-states Potts model free energies based on the Fortuin-Kasteleyn
representation of the model. With this procedure, we compute directly the
ordered phase partition function up to order 10 in 1/sqrt{q}. The energy
cumulants at the transition can be obtained with suitable resummation and come
out large for q less or around 15. As a consequence, expansions of the free
energies around the transition temperature are useless for not large enough
values of q. In particular the pure phase specific heats are predicted to be
much larger, at q < 15, than the values extracted from current finite size
scaling analysis of extrema, whereas they agree very well with recent values
extracted at the transition point.
