  The Bose-Hubbard model with an effective off-site three-body tunneling,
characterized by jumps towards one another, between one atom on a site and a
pair atoms on the neighborhood site, is studied systematically on a
one-dimensional lattice, by using the density matrix renormalization group
method. The off-site trimer superfluid, condensing at momentum $k=0$, emerges
in the softcore Bose-Hubbard model but it disappears in the hardcore
Bose-Hubbard model. Our results numerically verify that the off-site trimer
superfluid phase derived in the momentum space from [Phys. Rev. A {\bf 81},
011601(R) (2010)] is stable in the thermodynamic limit. The off-site trimer
superfluid phase, the partially off-site trimer superfluid phase and the Mott
insulator phase are found, as well as interesting phase transitions, such as
the continuous or first-order phase transition from the trimer superfluid phase
to the Mott insulator phase. Our results are helpful in realizing this novel
off-site trimer superfluid phase by cold atom experiments.
