  We report a $^{139}$La-NMR study of polycrystalline samples of
multi($n$)-layered nickelates, La$_3$Ni$_2$O$_{7-\delta}$ ($n=2$) and
La$_4$Ni$_3$O$_{10-\delta}$ ($n=3$), at ambient pressure. Measurements of the
nuclear magnetic resonance (NMR) spectra and nuclear spin relaxation rate
($1/T_1$) indicate the emergence of a density wave order with a gap below
$T^*\sim150$ K for La$_3$Ni$_2$O$_{7-\delta}$ and $\sim130$ K for
La$_4$Ni$_3$O$_{10-\delta}$. The finite value of $1/T_1$ below $T^*$ indicates
metallic ground states with the remaining density of states at the Fermi level
($E_{\rm F}$) under the density wave order. These features are attributed to
multiple $d$ electron bands with different characteristics. Above $T^*$, the
gradual decrease in $1/T_1T$ upon cooling implies the presence of a band with
flat dispersion near $E_{\rm F}$. From our microscopic probes, we point out
that these nickelates ($n=2$ and $3$) possess similar electronic states despite
the difference in the formal valence of the Ni-$d$ electron states, which
provides a basis for understanding the novel high-$T_{\rm c}$ superconductivity
under high pressures.
