Characterization of almost-Riordan arrays with row sums (Q6655727)

An almost-Riordan array represents a broader class of Riordan arrays, characterized by its flexibility and extended structural features. These arrays maintain the essential properties of Riordan arrays but incorporate modifications such as nonlinear transformations, shifts, scaling factors, or perturbations within their generating functions. Due to these enhancements, almost-Riordan arrays are widely applicable in combinatorics, graph theory, and coding theory, offering new perspectives on generating functions and recurrence relations.\N\NThe behavior of almost-Riordan arrays and their inverses is examined in this paper through generating functions associated with row sums, alternating row sums, and weighted row sums. The so-called \(A\), \(Z\), and \(\omega\)-sequences of these arrays are identified by analyzing these generating functions.\N\NFurthermore, the product of two almost-Riordan arrays is derived using the same generating function framework. This approach highlights the algebraic and combinatorial properties of almost-Riordan arrays, demonstrating their versatility in mathematical exploration.