  We study the possibility of superconductivity in a trilayer Ruddlesden-Popper
nickelate La$_4$Ni$_3$O$_{10}$ under pressure both theoretically and
experimentally, making comparison with the recently discovered high $T_c$
superconductor La$_3$Ni$_2$O$_7$, a bilayer nickelate. Through DFT
calculations, we find that a structural phase transition from monoclinic to
tetragonal takes place around 10 - 15 GPa. Using the tetragonal crystal
structure, we theoretically investigate the possibility of superconductivity,
where a combination of fluctuation exchange approximation and linearized
Eliashberg equation is applied to a six-orbital model constructed from first
principles band calculation. The obtained results suggests that
La$_4$Ni$_3$O$_{10}$ may also become superconducting under high pressure with
$T_c$ comparable to some cuprates, although it is not as high as
La$_3$Ni$_2$O$_7$. We also perform experimental studies using our
polycrystalline samples of La$_3$Ni$_2$O$_{7.01}$ and La$_4$Ni$_3$O$_{9.99}$.
The superconducting transition of La$_3$Ni$_2$O$_{7.01}$, with a maximum onset
$T_c$ of 67.0 K at a pressure of 26.5 GPa, is confirmed by a drop in the
electrical resistance, as well as the magnetic field dependence of the
resistance. Quite interestingly, similar temperature and magnetic field
dependencies of the resistance are observed also for La$_4$Ni$_3$O$_{9.99}$,
where a drop in the resistance is observed at lower temperatures compared to
La$_3$Ni$_2$O$_{7.01}$, under pressures of 32.8 GPa and above. Given the
theoretical expectation, the reduction in the resistance can most likely be
attributed to the occurrence of superconductivity in La$_4$Ni$_3$O$_{9.99}$.
The temperature at which the resistance deviates from a linear behavior,
considered as the onset $T_c$, monotonically increases up to 23 K at 79.2 GPa,
which is opposite to the pressure dependence of $T_c$ in La3Ni2O7.01.
