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.




fcn binomial(n,k){ (1).reduce(k,fcn(p,i,n){ p*(n-i+1)/i },1,n) }
fcn pascal_upp(n){ [[(i,j); n; n; '{ binomial(j,i) }]]:toMatrix(_) } // [[..]] is list comprehension
fcn pascal_low(n){ [[(i,j); n; n; binomial]]:toMatrix(_) }
fcn pascal_sym(n){ [[(i,j); n; n; '{ binomial(i+j,i) }]]:toMatrix(_) }
fcn toMatrix(ns){ // turn a string of numbers into a square matrix (list of lists)
   cols:=ns.len().toFloat().sqrt().toInt();
   ns.pump(List,T(Void.Read,cols-1),List.create)
}