  The central kiloparsec region of the Andromeda galaxy is relatively gas poor,
while the interstellar medium appears to be concentrated in a ring-like
structure at about 10 kpc radius. The central gas depletion has been attributed
to a possible head-on collision 200 Myr ago, supported by the existence of an
offset inner ring of warm dust. We present new IRAM 30m radio telescope
observations of the molecular gas in the central region, and the detection of
CO and its isotopes $^{13}$CO(2-1) and C$^{18}$O(2-1), together with the dense
gas tracers, HCN(1-0) and HCO+(1-0). A systematic study of the observed peak
temperatures with non-LTE equilibrium simulations shows that the detected lines
trace dense regions with n$_{H_2}$ in the range 2.5 $10^4 - 5.6 10^5$
cm$^{-3}$, while the gas is very clumpy with a beam filling factor of 0.5-2
10$^{-2}$. This is compatible with the dust mass derived from the far-infrared
emission, assuming a dust-to-gas mass ratio of 0.01 with a typical clump size
of 2 pc. We also show that the gas is optically thin in all lines except for
$^{12}$CO(1-0) and $^{12}$CO(2-1), CO lines are close to their thermal
equilibrium condition at 17.5-20 K, the molecular hydrogen density is larger
than critical and HCN and HCO+ lines have a subthermal excitation temperature
of 9 K with a density smaller than critical. The average $^{12}$CO/$^{13}$CO
line ratio is high (~21), and close to the $^{12}$CO/C$^{18}$O ratio (~30) that
was measured in the north-western region and estimated in the south-east
stacking. The fact that the optically thin $^{13}$CO and C$^{18}$O lines have
comparable intensities means that the secondary element $^{13}$C is depleted
with respect to the primary $^{12}$C, as is expected just after a recent star
formation. This suggests that there has been a recent starburst in the central
region, supporting the head-on collision scenario.
