New 12CO J=4-3 and 13CO J=3-2 observations of the N159 region in the Large Magellanic Cloud have been made. The 12CO J=4-3 distribution is separated into three clumps. These new measurements toward the three clumps are used in coupled calculations of molecular rotational excitation and line radiation transfer, along with other transitions of the 12CO as well as the isotope transitions of 13CO. The temperatures and densities are determined to be ~70-80K and ~3x10^3 cm-3 in N159W and N159E and ~30K and ~1.6x10^3 cm-3 in N159S. These results are compared with the star formation activity. The N159E clump is associated with embedded cluster(s) as observed at 24 micron and the derived high temperature is explained as due to the heating by these sources. The N159E clump is likely responsible for a dark lane in a large HII region by the dust extinction. The N159W clump is associated with embedded clusters mainly toward the eastern edge of the clump only. These clusters show offsets of 20-40 from the 12CO J=4-3 peak and are probably responsible for heating indicated by the derived high temperature. The N159W clump exhibits no sign of star formation toward the 12CO J=4-3 peak position and its western region. We suggest that the N159W peak represents a pre-star-cluster core of ~105M_sol which deserves further detailed studies. Note that recent star formation took place between N159W and N159E as indicated by several star clusters and HII regions, while the natal molecular gas toward the stars have already been dissipated by the ionization and stellar winds of the OB stars. The N159S clump shows little sign of star formation as is consistent with the lower temperature and somewhat lower density. The N159S clump is also a candidate for future star formation.
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