The series of nickel-oxide compounds LaNiO3 (formal Ni(d 7)), La2NiO4 (formal Ni(d 8)) and LaNiO2 (formal Ni(d 9)) is investigated by first-principles many-body theory, using a combination of density functional theory, self-interaction correction and dynamical mean-field theory. The characteristics of these different nickelates, in good agreement with available experimental data, is revealed by employing a compound-independent choice for the local Coulomb interactions. The dichotomy within the low-energy dominant Ni-e g sector of {dz2,dx2-y2} kind is rising with growing Ni(3d) filling across the series. An intermediate-coupling scheme for spin-polarized calculations is introduced, which leads to very weak Ni ordered moments for the infinite-layer compound LaNiO2 in contrast to the robust-moment system La2NiO4.