We have performed a systematic ground state geometry search for the singly charged Si-n cations in the medium-size range (n less than or equal to 20) using density functional theory in the local density approximation (LDA) and generalized gradient approximation (GGA). The structures resulting for n less than or equal to 18 generally follow the prolate ＇＇stacked Si-9 tricapped trigonal prism＇＇ pattern recently established for the lowest energy geometries of neutral silicon clusters in this size range. However, the global minima of Si-n and Si-n(+) for n = 6, 8, 11, 12, and 13 differ significantly in their details. For S-19 and Si-20 neutrals and cations, GGA renders the prolate stacks practically isoenergetic with the near-spherical structures that are global minima in LDA. The mobilities in He gas evaluated for all lowest energy Si-n(+) geometries using the trajectory method agree with the experiment, except for n=18 where the second lowest isomer fits the measurements. The effect of gradient corrections for either the neutral or cationic clusters is subtle, but their inclusion proves to be critical for obtaining agreement with the mobility measurements in the n=15-20 range. We have also determined ionization potentials for our Si-n neutral geometries and found that all experimental size-dependent trends are reproduced for n less than or equal to 19. This particularly supports our structural assignments for Si-9, Si-11, Si-12, and Si-17 neutrals. The good overall agreement between the measured and calculated properties supports the elucidation of the "prolate" family of silicon clusters as stacks of trigonal prisms.