We calculate the evolution of massive stars , which undergo pulsational pair-instability ( PPI ) when the O-rich core is formed .
The evolution from the main-sequence through the onset of PPI is calculated for stars with the initial masses of 80 – 140 M _ { \odot } and metallicities of Z = 10 ^ { -3 } -1.0 Z _ { \odot } .
Because of mass loss , Z \leq 0.5 Z _ { \odot } is necessary for stars to form He cores massive enough ( i.e. , mass > 40 ~ { } M _ { \odot } ) to undergo PPI .
The hydrodynamical phase of evolution from PPI through the beginning of Fe core collapse is calculated for the He cores with masses of 40 - 62 ~ { } M _ { \odot } and Z = 0 .
During PPI , electron-positron pair production causes a rapid contraction of the O-rich core which triggers explosive O-burning and a pulsation of the core .
We study the mass dependence of the pulsation dynamics , thermodynamics , and nucleosynthesis .
The pulsations are stronger for more massive He cores and result in such a large amount of mass ejection such as 3 – 13 M _ { \odot } for 40 - 62 ~ { } M _ { \odot } He cores .
These He cores eventually undergo Fe-core collapse .
The 64 ~ { } M _ { \odot } He core undergoes complete disruption and becomes a pair-instability supernova .
The H-free circumstellar matter ejected around these He cores is massive enough for to explain the observed light curve of Type I ( H-free ) superluminous supernovae with circumstellar interaction .
We also note that the mass ejection sets the maximum mass of black holes ( BHs ) to be \sim 50 M _ { \odot } , which is consistent with the masses of BHs recently detected by VIRGO and aLIGO .