Rayleigh-Taylor instability (RTI) in cylindrical geometry initiated by velocity and interface perturbations is investigated analytically through a third-order weakly nonlinear (WN) model.When the initial velocity perturbation is comparable to the interface perturbation,the coupling between them plays a significant role.The difference between the RTI growth initiated only by a velocity perturbation and that only by an interface perturbation in the WN stage is negligibly small.The effects of the mode number on the first three harmonics are discussed respectively.The low-mode number perturbation leads to large amplitudes of RTI growth.The Atwood number and initial perturbation dependencies of the nonlinear saturation amplitude of the fundamental mode are analyzed clearly.When the mode number of the perturbation is large enough,the WN results in planar geometry are recovered.