Under certain conditions the cycle index of symmetric group sums are shown to be equivalent to limiting Jacobi Polynomials or hypergeometric equations of special form when z=1. These sums are found to be numbers from select element sequences such as the Padovan sequence. Expanding the space of symmetric groups by changes in two parameters g and j, allows for similar calculation of numbers from other classes of sequences. The common theme of these sequences is the coloration of objects within various symmetric groups. Sequence numbers are shown to be expansions of the cycle indices in powers of the constant coefficient of a class of polynomials of odd order. Special non-linear recurrences occur when g is or j is an even integer.