Using angle-resolved photoemission, we have observed sharp quasiparticlelike peaks in the prototypical layered manganite La(2-2x)Sr(1+2x)Mn(2)O(7) (x=0.36,0.38). We focus on the (pi,0) regions of k space and study their electronic scattering rates and dispersion kinks, uncovering bilayer-split bands, the critical energy scales, momentum scales, and strengths of the interactions that renormalize the electrons. To identify these bosons, we measured phonon dispersions in the energy range of the kink by inelastic neutron scattering, finding a good match in both energy and momentum to the oxygen bond-stretching phonons.