  To evaluate the Hadamard finite-part integrals accurately, a novel
interpolatory-type quadrature is proposed in this article. In our approach,
numerical divided difference is utilized to represent the high order
derivatives of the integrated function, which make it possible to reduced the
numerical quadrature into a concise formula based on the cycle index for
symmetric group. In addition, convergence analysis is presented and the error
estimation is given. Numerical results are presented on cases with different
weight functions, which substantiate the performance of the proposed method.
