  Present time-domain astronomy efforts will unveil a variety of rare
transients. We focus here on pulsational pair-instability evolution, which can
result in signatures observable with electromagnetic and gravitational waves.
We simulate grids of bare helium stars to characterize the resulting black hole
(BH) masses and ejecta composition, velocity, and thermal state. The stars do
not react "elastically" to the thermonuclear explosion: there is not a
one-to-one correspondence between pair-instability driven ignition and mass
ejections, causing ambiguity in what is an observable pulse. In agreement with
previous studies, we find that for carbon-oxygen core masses 28Msun<
M_CO<30.5Msun the explosions are not strong enough to affect the surface. With
increasing mass, they first cause large radial expansion
(30.5Msun<M_CO<31.4Msun), and finally, also mass ejection episodes
(M_CO>31.4Msun). The lowest mass to be fully disrupted in a pair-instability
supernova is M_CO=57Msun. Models with M_CO>121Msun reach the
photodisintegration regime, resulting in BHs with M_BH>125Msun. If the
pulsating models produce BHs via (weak) explosions, the previously-ejected
material might be hit by the blast wave. We characterize the H-free
circumstellar material from the pulsational pair-instability of helium cores
assuming simply that the ejecta maintain a constant velocity after ejection.
Our models produce He-rich ejecta with mass 10^{-3}Msun<M_CSM<40Msun. These
ejecta are typically launched at a few thousand \kms and reach distances of
~10^{12}-10^{15} cm before core-collapse. The delays between mass ejection
events and the final collapse span a wide and mass-dependent range (from
sub-hour to 10^4 years), and the shells ejected can also collide with each
other. The range of properties we find suggests a possible connection with
(some) type Ibn supernovae.
