Growing evidence suggests that the first generation of stars may have been quite massive (~100-300 Msolar). Could these stars have left a distinct nucleosynthetic signature? We explore the nucleosynthesis of helium cores in the mass range MHe=64-133 Msolar, corresponding to main-sequence star masses of approximately 140-260 Msolar. Above MHe=133 Msolar, without rotation and using current reaction rates, a black hole is formed, and no nucleosynthesis is ejected. For lighter helium core masses, ~40-63 Msolar, violent pulsations occur, induced by the pair instability and accompanied by supernova-like mass ejection, but the star eventually produces a large iron core in hydrostatic equilibrium. It is likely that this core, too, collapses to a black hole, thus cleanly separating the heavy-element nucleosynthesis of pair instability supernovae from those of other masses, both above and below. Indeed, black hole formation is a likely outcome for all Population III stars with main-sequence masses between about 25 and 140 Msolar (MHe=9-63 Msolar) as well as those above 260 Msolar. Nucleosynthesis in pair instability supernovae varies greatly with the mass of the helium core. This core determines the maximum temperature reached during the bounce. At the upper range of exploding core masses, a maximum of 57 Msolar of 56Ni is produced, making these the most energetic and the brightest thermonuclear explosions in the universe. Integrating over a distribution of masses, we find that pair instability supernovae produce a roughly solar distribution of nuclei having even nuclear charge (Si, S, Ar, etc.) but are remarkably deficient in producing elements with odd nuclear charge-Na, Al, P, V, Mn, etc. This is because there is no stage of stable post-helium burning to set the neutron excess. Also, essentially no elements heavier than zinc are produced owing to a lack of s- and r-processes. The Fe/Si ratio is quite sensitive to whether the upper bound on the initial mass function is over 260 Msolar or somewhere between 140 and 260 Msolar. When the yields of pair instability supernovae are combined with reasonable estimates of the nucleosynthesis of Population III stars from 12 to 40 Msolar, this distinctive pattern of deficient production of odd-Z elements persists. Some possible strategies for testing our predictions are discussed.