Numbers Intricacy Investigation

All foreseeable approaches to derive polynomial expressions for the power-sum of the natural sequence are presented here. Throughout binomial coefficients play as the key role of linking together the product-sums and the power-sums. The author takes the opportunity to sort out the intricate liaisons among Stirling numbers of both kinds Euler numbers of two orders ordered and Eulerian Bell polynomials. He further generalizes the related numbers based on the natural sequence to those that are arithmetically progressive sequence based so that various structures of triangular arrays can be built on top of different underlying bases.