  General expressions for the R\'enyi entropies and central charges for higher
derivative free spinors and scalars on even spheres are obtained using a direct
spectral method on a compact lune division of the sphere. Formulae and numbers
are rapidly obtained for any dimension and order of derivative.
  The relation between the conformal anomaly and the hyperbolic free energy is
briefly explored using standard expansions.
  A field theoretic derivation of the central charge formula for higher
derivative scalars in any (even) dimension, given by Osborn and Stergiou and by
Gliozzi {\it et al}, is thereby provided. The corresponding extension to
spinors is made.
  Generalised Bernoulli polynomials play an important technical role.
