  In this study, we perform a numerical simulation on the recently discovered
high-temperature superconductor ($T_c$= 73K) Ba$_2$CuO$_{3.2}$ \cite{lietal}
while focusing on doping dependence of alternating CuO$_6$ octahedra and CuO
chain-like states. Employing the multiband random-phase approximation, we
compute the spin-fluctuation mediated pairing interaction, subsequently
determining its pairing eigenvalues and eigenfunctions relative to
oxygen-doping levels. We find that, for the certain range of hole doping in
Ba$_2$CuO$_{3+\delta}$, a singlet $d_{x^2-y^2}$-wave pairing symmetry emerges
as long as we keep the doping below the critical value $x_{c}$. Interestingly
upon hole doping, the dominant pairing symmetry undergoes a transition to a
triplet (odd paring) type from the singlet state. This change in pairing is
driven by the competition between the nesting vectors coming from the Fermi
surface of $d_{z^2}$ and $d_{x^2-y^2}$ orbitals within the CuO$_6$ octahedra.
This triplet state is attainable through hole doping, while supressing
inter-layer self-doping effects. Furthermore, we present the density of states
within the superconducting phase, offering a potential comparison with
tunnelling spectra in Ba$_2$CuO$_{3+\delta}$. Our research provides novel
insights into the intricate pairing symmetries in Ba$_2$CuO$_{3+\delta}$ and
their underlying pairing mechanisms.
