The Keçeci Binomial Square: A Reinterpretation of the Standard Binomial Expansion and Its Potential Applications

The Keçeci Binomial Square: A Reinterpretation of the Standard Binomial Expansion and Its Potential Applications Mehmet Keçeci1 1ORCID: https://orcid.org/0000-0001-9937-9839, İstanbul, Türkiye Received: 15.05.2025 Abstract: The Khayyam–Pascal Triangle (also known as the Binomial Triangle) a fundamental structure in mathematical combinatorics and number theory, powerfully visualizes the regular and predictable distribution of binomial coefficients. This study introduces an innovative approach termed the "Keçeci Binomial Square" (KBS, (first defined: March 2025)), which defines numerical series within Pascal's Triangle characterized by specific geometric and structural properties. Rather than directly manipulating the standard binomial expansion, KBS focuses on a specialized selection and analysis of Pascal's Triangle elements that constitute the coefficients of these expansions. The core definition of KBS relies on selecting an N x N square region from Pascal's Triangle. This selection is dynamically determined by a user-specified start_row_index and an alignment_type ("left", "right", "center") that dictates how coefficients are positioned within each row. The resulting numerical series comprises N segments of N elements each, drawn from consecutive rows of Pascal's Triangle. The sum of these elements constitutes one of the primary outputs of the KBS. This structure offers an opportunity to examine not only the individual values of binomial coefficients but also their behavior within a particular regional integration. The academic value of the KBS concept lies in its ability to facilitate the discovery of local patterns and relationships within Pascal's Triangle. This approach not only aids in visualizing binomial coefficients and combinatorial principles in mathematical education but also provides a framework for investigating specific additive properties in number theory or particular cases in algorithmic analysis. For instance, the impact of different alignment types on the selected series sums, or potential connections between KBS series for specific N values and other known number sequences (e.g., Fibonacci, Catalan), present fertile grounds for future research. In conclusion, the Keçeci Binomial Square offers a systematic method for re-contextualizing a well-established mathematical structure, thereby revealing hidden relationships and additive properties among binomial coefficients. This framework holds the potential to stimulate new research in theoretical mathematics and offer novel perspectives in applied fields (e.g., combinatorial optimization, data analysis). Future work can delve into a deeper mathematical analysis of different KBS configurations and test their practical implications across various disciplines. Keywords: Khayyam's Triangle, Pascal's Triangle, Binomial Triangle, Binomial Coefficients, Number Series, Square Selection, Combinatorics, Pattern Recognition, Number Theory, Visualization, Keçeci Binomial Square, Keçeci Square, Keçeci Binom Karesi, Keçeci Karesi