A pascal matrix is a two-dimensional square matrix holding numbers from   Pascal's triangle,   also known as   binomial coefficients   and which can be shown as   nCr.

Shown below are truncated   5-by-5   matrices   M[i, j]   for   i,j   in range   0..4. 


A Pascal upper-triangular matrix that is populated with   jCi:

[[1, 1, 1, 1, 1],
 [0, 1, 2, 3, 4],
 [0, 0, 1, 3, 6],
 [0, 0, 0, 1, 4],
 [0, 0, 0, 0, 1]]

A Pascal lower-triangular matrix that is populated with   iCj   (the transpose of the upper-triangular matrix):

[[1, 0, 0, 0, 0],
 [1, 1, 0, 0, 0],
 [1, 2, 1, 0, 0],
 [1, 3, 3, 1, 0],
 [1, 4, 6, 4, 1]]

A Pascal symmetric matrix that is populated with   i+jCi:

[[1, 1, 1, 1, 1],
 [1, 2, 3, 4, 5],
 [1, 3, 6, 10, 15],
 [1, 4, 10, 20, 35],
 [1, 5, 15, 35, 70]]



Task

Write functions capable of generating each of the three forms of   n-by-n   matrices.

Use those functions to display upper, lower, and symmetric Pascal   5-by-5   matrices on this page.

The output should distinguish between different matrices and the rows of each matrix   (no showing a list of 25 numbers assuming the reader should split it into rows).



Note
 
The   Cholesky decomposition   of a Pascal symmetric matrix is the Pascal lower-triangle matrix of the same size.




lower.pascal <- function(n) {
  a <- matrix(0, n, n)
  a[, 1] <- 1
  if (n > 1) {
    for (i in 2:n) {
      j <- 2:i
      a[i, j] <- a[i - 1, j - 1] + a[i - 1, j]
    }
  }
  a
}
 
# Alternate version
lower.pascal.alt <- function(n) {
  a <- matrix(0, n, n)
  a[, 1] <- 1
  if (n > 1) {
    for (j in 2:n) {
      i <- j:n
      a[i, j] <- cumsum(a[i - 1, j - 1])
    }
  }
  a
}
 
# While it's possible to modify lower.pascal to get the upper matrix,
# here we simply transpose the lower one.
upper.pascal <- function(n) t(lower.pascal(n))
 
symm.pascal <- function(n) {
  a <- matrix(0, n, n)
  a[, 1] <- 1
  for (i in 2:n) {
    a[, i] <- cumsum(a[, i - 1])
  }
  a
}