Extended Central Factorial Numbers and the Flickering Operator

TL;DR

This paper introduces the flickering operator, a new recursive framework for extended central factorial numbers that unifies various combinatorial sequences and offers an integer-based power sum expansion.

Contribution

The study presents a novel flickering operator that unifies multiple combinatorial sequences into a single recursive framework, avoiding fractional Bernoulli numbers.

Findings

Provides explicit closed-form expressions for the extended central factorial numbers.

Demonstrates the flickering operator's ability to generate complete integer sequences.

Offers a Python implementation of the proposed framework.

Abstract

This paper introduces a class of extended central factorial numbers generated by a parity-dependent recurrence relation, termed the "flickering operator". We demonstrate that the resulting triangular structure, now indexed as OEIS A395021, provides a unified recursive framework for alternating bit sequences (A000975) and normalized tangent-secant coefficients (A036969). This study provides an alternative integer-based expansion for power sums. While similar to the central factorial methods explored by Knuth (1993), our flickering basis offers an integrated computational scheme that avoids fractional Bernoulli numbers by construction. We provide explicit closed-form expressions, discuss its geometric derivation from finite difference tables, and present a full Python implementation. Structural Synthesis. A key contribution of this work is the unification of previously disparate combinatorial sequences into a single coherent framework. While certain columns of the flickering triangle T(n, k) (such as A008957) could be partially retrieved from the diagonals of existing central factorial arrays, our structure provides a complete representation including previously unindexed even-positioned terms. Furthermore, the row-wise analysis reveals that the flickering operator generates full integer sequences where previously only the odd-indexed elements (e.g., A002451) were identified. This synthesis bridges the gap between these sequences, positioning A395021 as the underlying master structure.