  We study a generalization of the D-dimensional Vasiliev theory to include a
tower of partially massless fields. This theory is obtained by replacing the
usual higher-spin algebra of Killing tensors on (A)dS with a generalization
that includes "third-order" Killing tensors. Gauging this algebra with the
Vasiliev formalism leads to a fully non-linear theory which is expected to be
UV complete, includes gravity, and can live on dS as well as AdS. The
linearized spectrum includes three massive particles and an infinite tower of
partially massless particles, in addition to the usual spectrum of particles
present in the Vasiliev theory, in agreement with predictions from a putative
dual CFT with the same symmetry algebra. We compute the masses of the particles
which are not fixed by the massless or partially massless gauge symmetry,
finding precise agreement with the CFT predictions. This involves computing
several dozen of the lowest-lying terms in the expansion of the trilinear form
of the enlarged higher-spin algebra. We also discuss nuances in the theory that
occur in specific dimensions; in particular, the theory dramatically truncates
in bulk dimensions D=3,5 and has non-diagonalizable mixings which occur in
D=4,7.
