To investigate the origin of aberration coefficients and their mutual relations in charged particle optics, the high-order coefficients are studied using both semi-analytical and differential algebraic methods. These coefficients have been calculated up to the 11th order for various distributions of a rotationally symmetric magnetic field. The computation errors in the results of both methods are decreased to at least less than 10 -12. Since the contribution of each term in the calculation heavily depends on the field strength, the formation of spherical-type aberrations is studied by selecting the effective terms. Some relations among the coefficients are found when the field and the paraxial trajectory are symmetric with respect to the principal plane of a lens.