The partial Bell Polynomial expresses polynomials of any order and is convenient to calculate parent sequences and convolutions. The Perrin sequence is used as an example showing that a convolution hierarchy of sequences of various order can be algebraically equal. The condition for algebraic equality is shown and how the calculus (derivatives) of these convolutions are determined. Linear mappings of the Parent and Element sequences are shown to be group homomorphisms.
Sequence entanglement of the element sequences is described. Entanglement is possible when the three roots of the discriminant equation are real.
Algebra of Integer Sequences and their Convolution using Partial Bell Polynomials_
This chalkboard discusses the previously derived hypergeometric equation for calculating Perrin numbers. The equation is also found to calculate the number of k-th order maximal independent sets in a cycle graph. Convolution sequences are found to be related to the k-th order numbers and to two combinatorial numbers.
Perrin Numbers, a Hypergeometic Function and Convolution Sequences
A paradox is found between derivatives of the ISPs and discrete convolution of integer sequences. ISP derivatives are convolution operators!