  We generalize the technique of linked cluster expansions on hypercubic
lattices to actions that couple fields at lattice sites which are not nearest
neighbours. We show that in this case the graphical expansion can be arranged
in such a way that the classes of graphs to be considered are identical to
those of the pure nearest neighbour interaction. The only change then concerns
the computation of lattice imbedding numbers. All the complications that arise
can be reduced to a generalization of the notion of free random walks,
including hopping beyond nearest neighbour. Explicit expressions for
combinatorical numbers of the latter are given. We show that under some general
conditions the linked cluster expansion series have a non-vanishing radius of
convergence.
