  The purpose of this article is to provide a solution to the $m$-fold Laplace
equation in the half space $R_+^d$ under certain Dirichlet conditions. The
solutions we present are a series of $m$ boundary layer potentials. We give
explicit formulas for these layer potentials as linear combinations of powers
of the Laplacian applied to the Dirichlet data, with coefficients determined by
certain path counting problems.
