  One of the best understood families of logarithmic conformal field theories
is that consisting of the (1,p) models (p = 2, 3, ...) of central charge
c_{1,p} = 1 - 6 (p-1)^2 / p. This family includes the theories corresponding to
the singlet algebras M(p) and the triplet algebras W(p), as well as the
ubiquitous symplectic fermions theory. In this work, these algebras are
realized through a coset construction.
  The W^(2)_n algebra of level k was introduced by Feigin and Semikhatov as a
(conjectured) quantum hamiltonian reduction of affine sl(n)_k, generalising the
Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3,
vertex algebras B_p are constructed as subalgebras of the kernel of certain
screening charges acting on a rank 2 lattice vertex algebra of indefinite
signature. It is shown that for p <= 5, the algebra B_p is a homomorphic image
of W^(2)_{p-1} at level -(p-1)^2 / p and that the known part of the operator
product algebra of the latter algebra is consistent with this holding for p>5
as well. The triplet algebra W(p) is then realised as a coset inside the full
kernel of the screening operator, while the singlet algebra M(p) is similarly
realised inside B_p. As an application, and to illustrate these results, the
coset character decompositions are explicitly worked out for p=2 and 3.
