A new version of the standard series expansion algorithm for calculation of one-center many-particle correlation integrals is proposed. The performance of this algorithm is considerably increased due to a special order of computation of auxiliary $W$-integrals using recurrent formulas to express one integral in terms of another integral. This technique allows one to compute an array of $W$-functions and to retain the numerical accuracy of computations by evaluating a small number of its components with the use of infinite series. Evaluation of $W$-integrals by infinite series and extrapolation procedures are also improved.

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