  We point out the joint occurrence of Pascal triangle patterns and power-law
scaling in the standard logistic map, or more generally, in unimodal maps. It
is known that these features are present in its two types of bifurcation
cascades: period and chaotic- band doubling of attractors. Approximate Pascal
triangles are exhibited by the sets of lengths of supercycle diameters and by
the sets of widths of opening bands. Additionally, power-law scaling manifests
along periodic attractor supercycle positions and chaotic band splitting
points. Consequently, the attractor at the mutual accumulation point of the
doubling cascades, the onset of chaos, displays both Gaussian and power-law
distributions. Their combined existence implies both ordinary and exceptional
statistical-mechanical descriptions of dynamical properties.
