Investigating higher-dimensional Pascal-type structures and their implications, with a structured exploration of how monomial powers generate factorial coefficients and vanish after finite layers under repeated transformations.

The forward difference operator and its relationship to binomial coefficients are well-established in discrete mathematics. However, the explicit organization of iterated finite differences (∆(ℓ) p) as a family of layered Pascal-type triangular arrays has not been systematically developed in the literature. We present a unified framework in which each layer index ℓ produces a distinct coefficient triangle: the classical Pascal triangle emerges as Layer 0, while higher layers yield transformed structures that retain Pascal-like organization. We derive a closed-form coefficient generator Cp,ℓ(v) indexed by three parameters, power, layer, and position, that directly produces individual polynomial coefficients without requiring polynomial expansion or basis conversion through Stirling numbers. This formula enables O(ℓ) access to specific coefficients in the layered structure, providing computational advantages for applications requiring sparse coefficient extraction when p ≫ ℓ. The framework reveals that several previously disconnected OEIS sequences, including A259569 (second-order differences), A001117, A000918, A000919, and A000920, are fixed-layer projections of a single unified structure. The entire construction is developed using only elementary combinatorial tools accessible at the undergraduate level, making it suitable for both computational applications and pedagogical exposition.