Hankel transform of a series reversion of a certain rational function

Starting from a sequence (vn)n∈N0, defined by the series reversion of a certain rational function B(x), we use the method based on orthogonal polynomials to give closed-form evaluation of the Hankel transform of sequence (vn)n∈N0 and corresponding shifted sequences. That gives a proof the conjecture by Barry [4] concerning the ratio equalities connecting those terms. Moreover, in a special case we compute LU decomposition of the Hankel matrix of a shifted sequence , which reproves the expression for the Hankel transform of that sequence.