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The authors present a fast procedure for computing a "modified" triangular factorization of Hankel, quasi-Hankel (matrices congruent in a certain sense to Hankel matrices) and sign-modified quasi-Hankel (products of quasi-Hankel and signature matrices) matrices. A fast procedure for computing inverse of Hankel and quasi-Hankel matrices is also presented. A modified triangular factorization is an $LDL*$ factorization, where $L$ is lower triangular with unit diagonal entries and $D$ is a block diagonal matrix with possibly varying block sizes. Only matrices with all leading minors nonzero, often called strongly regular, will always have a purely diagonal and nonsingular $D$ matrix. The matrices studied in this paper have diagonal blocks with a particular Hankel-(like) structure. The algorithms presented here are obtained by extending a generating function approach of Lev-Ari and Kailath for matrices with a generalized displacement structure. A particular application of the results is a fast method of computing the rank profile and inertia of the matrices involved.