  We introduce a new approach that allows us to determine the structure of
Zhu's algebra for certain vertex operator (super)algebras which admit
horizontal $\mathbb{Z} $-grading. By using this method and an earlier
description of Zhu's algebra for the singlet W-algebra, we completely describe
the structure of Zhu's algebra for the triplet vertex algebra W(p). As a
consequence, we prove that Zhu's algebra A(W(p)) and the related Poisson
algebra P(W(p)) have the same dimension. We also completely describe Zhu's
algebras for the N=1 triplet vertex operator superalgebra SW(m). Moreover, we
obtain similar results for the c=0 triplet vertex algebra W_{2,3} important in
logarithmic conformal field theory.
  Because our approach is "internal" we had to employ several constant term
identities for purposes of getting right upper bounds on dimension of Zhu's
algebras.
