A systematic approach to matrix forms of the Pascal triangle: The twelve triangular matrix forms and relations

This work initiates a systematic investigation into the matrix forms of the Pascal triangle as mathematical objects in their own right. The present paper is especially devoted to the so-called G-matrices, i.e. the set of the twelve ( n + 1 ) × ( n + 1 ) triangular matrix forms that can be derived from the Pascal triangle expanded to the level n ( 2 ≤ n ∈ N ) . For n = 1 , the G-matrix set reduces to a set of four distinct matrices. The twelve G-matrices are defined and the classic Pascal recursion is reformulated for each of the twelve G-matrices. Three sets of matrix transformations are then introduced to highlight different relations between the twelve G-matrices and for generating them from appropriately chosen subsets.