Response of the wave packet of a one-dimensional Coulomb atom to an intense laser field is calculated using the symmetrized split operator fast Fourier method.The high-order harmonic generation (HHG) of the initial state separately being the ground and excited states is presented.When the hardness parameter α in the soft Coulomb potential V(x) =-1/(x2+α) is chosen to be small enough,the so-called hard Coulomb potential V(x) =-1/|x| can be obtained.It is Well knawn that the hard one-dimensional Coulomb atom has an unstable ground state with an energy eigenvalue of ～0.5 and it has no states corresponding to physical states in the true atoms,and has the first and second excited states being degenerate.The parity effects on the HHG can be seen from the first and second excited states of the hard one-dimensional Coulomb atom.The HHG spectra of the excited states from both the soft and hard Coulomb atom models are shown to have more complex structures and to be much stronger than the corresponding HHG spectrum of the ground state of the soft Coulomb model with α=2 in the same laser field.Laser-induced non-resonant one-photon emission is also observed.