  We mapped molecular gas toward the supernova remnant W44 in the HCO+ J=1-0
line with the Nobeyama Radio Observatory 45 m telescope and in the CO J=3-2
line with the Atacama Submillimeter Telescope Experiment 10 m telescope.
High-velocity emission wings were detected in both lines over the area where
the radio shell of W44 overlaps the molecular cloud in the plane of the sky. We
found that the average velocity distributions of the wing emission can be
fitted by a uniform expansion model. The best-fit expansion velocities are
12.2+-0.3 km/s and 13.2+-0.2 km/s in HCO+ and CO, respectively. The non-wing CO
J=3-2 component is also fitted by the same model with an expansion velocity of
4.7+-0.1 km/s . This component might be dominated by a post shock
higher-density region where the shock velocity had slowed down. The kinetic
energy of shocked molecular gas is estimated to be (3.5+-1.3)x10^{49} erg.
Adding this and the energy of the previously identified HI shell, we concluded
that (1.2+-0.2)x10^{50} erg has been converted into gas kinetic energy from the
initial baryonic energy of the W44 supernova. We also found ultra-high-velocity
CO J=3-2 wing emission with a velocity width of ~100 km/s at (l, b)=(+34.73d,
-0.47d). The origin of this extremely high-velocity wing is a mystery.
