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The partition function for the Anderson Hamiltonian is expanded in a power series of U, the Coulomb integral between d-electrons at an impurity site, and for the case in which electron-hole symmetry is maintained it is found that the odd-order terms vanish and the even-order terms can be described by the imaginary-time integrals of the fourth power of the Pfaffian constructed from the one-particle Green function. In the approximation that the Green function is replaced by its asymptotic form, almost all diagrams are cancelled and only bubble diagrams remain, and so one cannot avoid instability by this approximation.as RPA. Finally, for large U the perturbation expansion is formulated in