  We study the Hilbert series of a family of ideals J_\phi generated by powers
of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we
formulate this as a question about fatpoints in P^{n-1}. In the three variable
case this is equivalent to studying the dimension of a linear system on a blow
up of P^2. The ideals that arise have the points in very special position, but
because there are only seven points, we can apply results of Harbourne to
obtain the classes of the negative curves. Reducing to an effective, nef
divisor and using Riemann-Roch yields a formula for the Hilbert series. This
proves the n=3 case of a conjecture of Postnikov and Shapiro, which they later
showed true for all n. Postnikov and Shapiro observe that for a family of
ideals closely related to J_\phi a similar result often seems to hold, although
counterexamples exist for n=4 and n=5. Our methods allow us to prove that for
n=3 an analogous formula is indeed true. We close with a counterexample to a
conjecture Postnikov and Shapiro make about the minimal free resolution of
these ideals.
