  The goal of this paper is to study the dynamics of holomorphic
diffeomorphisms in C^n such that the resonances among the first 1<= r<= n
eigenvalues of the differential are generated over N by a finite number of
Q-linearly independent multi-indices (and more resonances are allowed for other
eigenvalues). We give sharp conditions for the existence of basins of
attraction where a Fatou coordinate can be defined. Furthermore, we obtain a
generalization of the Leau-Fatou flower theorem, providing a complete
description of the dynamics in a full neighborhood of the origin for 1-resonant
parabolically attracting holomorphic germs in Poincare'-Dulac normal form.
