Sequencean

Sequencean is a term used in theoretical discussions of discrete sequences to denote a derived sequence obtained by applying a fixed transformation to a base sequence, or more generally, a member of a family of sequences associated with that transformation.

In its common form, a sequencean S is obtained from a base sequence a = (a_n) by a linear recurrence of fixed order m, with coefficients that may depend on n: s_n = sum_{k=1}^m c_k(n) a_{n-k}. When the coefficients c_k(n) are simple, such as constants or polynomials in n, the resulting sequencean fits standard recurrence classes.

A widely used special case is the binomial transform, where s_n = sum_{k=0}^n binomial(n, k) a_k. This variant reshapes growth and is connected to generating functions, convolution, and other transform techniques commonly discussed in sequence analysis.

Properties of sequenceans include questions of convergence, growth rate, and stability under additional transforms. They are often studied via generating functions S(x) = sum s_n x^n and their relation to the generating function A(x) = sum a_n x^n.

Examples help illustrate the idea: if a_n = 1 for all n, the binomial transform yields s_n = 2^n. If a_n = n, then s_n = n 2^{n-1}. These cases highlight how the transformation influences asymptotics and combinatorial interpretations.

Related concepts include sequence space, generating function, binomial transform, and linear recurrence, which provide common tools for analyzing sequenceans and their properties.