In mathematics, particularly matrix theory and combinatorics, the symmetric Pascal matrix is a square matrix
from which you can derive binomial coefficients. The matrix is an order \(n\) symmetric
matrix with typical element given by \({S_{i,j}} = {{n!} \mathord{\left/
{\vphantom {{n!} {\left[ {r!\;\left( {n - r} \right)!} \right]}}} \right.
} {\left[ {r!\;\left( {n - r} \right)!} \right]}}\) where
\(n = i + j - 2\) and \(r = i - 1\). The binomial coefficients are elegantly recovered from the
symmetric Pascal matrix by performing an \(LU\) decomposition as \({\bf{S}} = {\bf{L}}\;{\bf{U}}\).
References
Call, G. S. and D. J. Velleman, (1993). Pascal's matrices, American Mathematical Monthly,
April 1993, 100, 372-376.
Edelman, A. and G. Strang, (2004). Pascal Matrices, American Mathematical Monthly,
111(3), 361-385.