  Spin excitations of layered copper oxide show various characteristic
features, depending on the carrier concentration. In this study, we conducted
inelastic neutron scattering (INS) measurements on La$_{2-x}$Sr$_{x}$CuO$_4$
(LSCO), with $x$ = 0, 0.075, 0.18, and 0.30 and La$_{2-x}$Sr$_{x}$NiO$_4$
(LSNO) with 1/3, to clarify the origin of the intensity enhancement in the
excitation spectrum of LSCO at the energy ($\omega$) of 16--19 meV [Phys. Rev.
B {\bf 91}, 224404 (2015), {\it ibid}. {\bf 93}, 094416 (2016)]. We confirmed
the presence of a peak-structure in the $\omega$-dependence of the local spin
susceptibility $\chi''(\omega)$ of superconducting (SC) LSCO with a peak energy
of 16--19 meV, where the spin excitations intersect optical phonon branches. A
comparable peak-structure is not observed in the insulating La$_2$CuO$_4$,
LSNO, and heavily overdoped LSCO with $x$ = 0.30. A dome-shaped $x$-dependence
of the integrated intensity around the peak energies is revealed for SC phase
by summarizing the present and previously reported results. Furthermore, our
phonon calculation on LCO shows the existence of two optical branches at
$\sim$19 meV that could stabilize stripe-alignment of carriers due to
out-of-plane vibrations of Cu or O of the CuO$_2$ planes. These results
indicate the interplay among spin, charge, and lattice dynamics and suggest
that the intensity enhancement is associated with their composite excitations.
