Let k be a field, \(X_ 1,...,X_ n\) indeterminates, \(R=k[X_ 1,...,X_ n]\) (graded by \(\deg (X_ i)=1)\) and I a homogeneous ideal of R. For a term ordering s, \(Lt_ s(I)\) is the leading term ideal associated to I with respect to s. It is known that the Hilbert- Poincaré series of R/I and \(R/Lt_ s(I)\) are the same - a result by F. S. Macaulay. Therefore the computation of \({\mathcal P}_{R/I}\) (the Hilbert-Poincaré series) is reduced to the computation of \({\mathcal P}_{R/I}\) for a monomial ideal I. The purpose of the paper is to prove a theorem which provides a formula for the computation of \({\mathcal P}_{R/I}\) (I monomial) via the computation of \({\mathcal P}_{S/J}\) \((S=k[X_ 1,...,\hat X_ i,...,X_ n]\supset J\) monomial). The complexity of the algorithm given by the theorem is estimated. It is in general of order \(n(I)^{2n-2}\), where n(I) is the number of a minimal generating set of I. The notion of Borel-normed ideals is defined, and it is shown that the complexity for a Borel-normed ideal I is of order \(n(I)^ 2\). The formula is applied to the computation of the multiplicity e(R/I) and the Hilbert-Poincaré series of a graded R- module.