We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincaré series converges at the critical exponent δ Γ. We obtain an explicit asymptotic for their orbital growth function. Namely, for any α ∈]1, 2[ and any slowly varying function L : R → (0, +∞), we construct N-dimensional Hadamard manifolds (X, g) of negative and pinched curvature, whose group of oriented isometries admits convergent geometrically finite subgroups Γ such that, as R → +∞, N Γ (R) := # {γ ∈ Γ ; d(o, γ · o) ≤ R} ∼ C Γ L(R) R α e δΓR , for some constant C Γ > 0.