  This is a continuation of arXiv:0908.4053, where, among other things, we
classified irreducible representations of the triplet vertex algebra W_{2,3}.
In this part we extend the classification to W_{2,p}, for all odd p>3. We also
determine the structure of the center of the Zhu algebra A(W_{2,p}) which
implies the existence of a family of logarithmic modules having L(0)-nilpotent
ranks 2 and 3. A logarithmic version of Macdonald-Morris constant term identity
plays a key role in the paper.
