We present new results on the microscopic nature of the ferroelectricity mechanisms in Ca3 Mn2O7 and Ca3Ti2O7. To the first approximation, we confirm the hybrid improper ferroelectric mechanism recently proposed by Benedek and Fennie for these Ruddlesden-Popper compounds. However, in Ca3Mn2O7 we find that there is a complex competition between lattice modes of different symmetry which leads to a phase coexistence over a large temperature range and the "symmetry trapping" of a soft mode. This trapping of the soft mode leads to a large uniaxial negative thermal expansion (NTE) reaching a maximum between 250 and 350 K (3.6×10^(-6)  K^{-1}) representing the only sizable NTE reported for these and related perovskite materials to date. Our results suggest a systematic strategy for designing and searching for ceramics with large NTE coefficients.