The partial Bell Polynomial is modified and shown to represent many integer sequences in the literature. The modified Bell Polynomial and its derivative are equivalent to Inter Sequence Polynomials and are applied to calculate Perrin and Padovan numbers, respectively. A detailed investigation of how these polynomials can reproduce the integer sequence is shown. The Bell Polynomial and number are important in combinatorics, partitions and mappings. This ability to describe various mappings with higher Bell Numbers provides a better understanding of the relation of integer sequences to discrete dynamic processes.
Tag: Perrin sequence
Chapter 34- Calculus of Integer Sequences
Pursue some path, however narrow and crooked, in which you can walk with love and reverence – Thoreau
All sequences representing monic cubic polynomials are shown to be generated by a single formula based on two modified Tribonacci sequences. These representations are multi-variable polynomials in x, y, and z and increase in the number of monomial terms with n. It is shown that these polynomials are continuous and can be integrated and differentiated. These inter-sequence polynomials (ISPs) obey the fundamental theorem of calculus and are graphically shown as surface sheets. Each sheet represents a set of sequences and are connected to the fundamental sequences described by Perrin, Lucas and Narayana and elemental repeated sequences. Sheets can be individual laminae or multiple sheets which may intersect other sheets.