  We study localized modes (LMs) of the one-dimensional
Gross-Pitaevskii/nonlinear Schr\"{o}dinger equation with a harmonic-oscillator
(parabolic) confining potential, and a periodically modulated coefficient in
front of the cubic term (nonlinear lattice pseudopotential). The equation
applies to a cigar-shaped Bose-Einstein condensate loaded in the combination of
a magnetic trap and an optical lattice which induces the periodic
pseudopotential via the Feshbach resonance. Families of stable LMs in the model
feature specific properties which result from the interplay between spatial
scales introduced by the parabolic trap and the period of the nonlinear
pseudopotential. Asymptotic results on the shapes and stability of LMs are
obtained for small-amplitude solutions and in the limit of a rapidly
oscillating nonlinear pseudopotential. We show that the presence of the lattice
pseudopotential may result in: (i) creation of new LM families which have no
counterparts in the case of the uniform nonlinearity; (ii) stabilization of
some previously unstable LM species; (iii) evolution of unstable LMs into a
pulsating mode trapped in one well of the lattice pseudopotential.
