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\name{symmetric.pascal.matrix}
\alias{symmetric.pascal.matrix}
\title{ Symmetric Pascal matrix }
\description{
This function returns an n by n symmetric Pascal matrix.
}
\usage{
symmetric.pascal.matrix(n)
}
\arguments{
\item{n}{ Order of the matrix }
}
\details{
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 \eqn{n} symmetric
matrix with typical element given by \eqn{{S_{i,j}} = {{n!} \mathord{\left/
{\vphantom {{n!} {\left[ {r!\;\left( {n - r} \right)!} \right]}}} \right.
} {\left[ {r!\;\left( {n - r} \right)!} \right]}}} where
\eqn{n = i + j - 2} and \eqn{r = i - 1}. The binomial coefficients are elegantly recovered from the
symmetric Pascal matrix by performing an \eqn{LU} decomposition as \eqn{{\bf{S}} = {\bf{L}}\;{\bf{U}}}.
}
\value{
An order \eqn{n} matrix.
}
\references{
Call, G. S. and D. J. Velleman, (1993). Pascal's matrices, \emph{American Mathematical Monthly},
April 1993, 100, 372-376.
Edelman, A. and G. Strang, (2004). Pascal Matrices, \emph{American Mathematical Monthly},
111(3), 361-385.
}
\author{ Frederick Novomestky \email{fnovomes@poly.edu} }
\note{
If the argument n is not a positive integer, the function presents an error message and stops.
}
\examples{
S <- symmetric.pascal.matrix( 4 )
print( S )
}
\keyword{ math }
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