  Within the framework of density functional theory, we present a study of
approximations to the enhancement factor of the non-interacting kinetic energy
functional $T_s[\rho]$. For this purpose, we employ the model of Liu and Parr
[S. Liu and R.G. Parr, Phys. Rev. A {\bf 55}, 1792 (1997)] based on a series
expansion of $T_s[\rho]$ involving powers of the density. Applications to 34
atoms, at the Hartree-Fock level showed that the enhancement factors present
peaks that are in excellent agreement with those of the exact ones and give an
accurate description of the shell structure of these atoms. The application of
Z-dependent expansions to represent some of the terms of these approximation
for neutral atoms and for positive and negative ions, which allows $T_s[\rho]$
to be cast in a very simple form, is also explored. Indications are given as to
how these functionals may be applied to molecules and clusters
