In the hypercomplex setting, the construction of hyperholomorphic polynomials is far from straightforward, since the product of two hyperholomorphic functions is, in general, not hyperholomorphic. In this talk, we present a family of hyperholomorphic polynomials that generalizes the classical complex powers within the framework of Clifford analysis. We investigate the structure of these polynomials and provide a characterization of their coefficients. Remarkably, the coefficients exhibit rich combinatorial patterns and are closely connected to special rational sequences, such as the Vietoris and Wigner numbers. These connections shed new light on the interplay between hypercomplex function theory and combinatorics, and suggest potential applications to the theory of special functions and discrete mathematics.