  The series of nickel-oxide compounds LaNiO$_3$ (formal Ni$(d^7)$),
La$_2$NiO$_4$ (formal Ni$(d^8)$) and LaNiO$_2$ (formal Ni$(d^9)$) is
investigated by first-principles many-body, 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 $\{d_{z^2},d_{x^2-y^2}\}$ 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 LaNiO$_2$ in contrast to the
robust-moment system La$_2$NiO$_4$.
