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.




function pascal_upper(integer n)
    sequence res = repeat(repeat(0,n),n)
    res[1] = repeat(1,n)
    for i=2 to n do
        for j=2 to i do
            res[j,i] = res[j,i-1]+res[j-1,i-1]
        end for
    end for
    return res
end function
 
function pascal_lower(integer n)
    sequence res = repeat(repeat(0,n),n)
    for i=1 to n do
        res[i,1] = 1
    end for
    for i=2 to n do
        for j=2 to i do
            res[i,j] = res[i-1,j]+res[i-1,j-1]
        end for
    end for
    return res
end function
 
function pascal_symmetric(integer n)
    sequence res = repeat(repeat(0,n),n)
    for i=1 to n do
        res[i,1] = 1
        res[1,i] = 1
    end for
    for i=2 to n do
        for j = 2 to n do
            res[i,j] = res[i-1,j]+res[i,j-1]
        end for
    end for
    return res
end function
 
ppOpt({pp_Nest,1,pp_IntCh,false,pp_IntFmt,"%2d"})
puts(1,"=== Pascal upper matrix ===\n")
pp(pascal_upper(5))
puts(1,"=== Pascal lower matrix ===\n")
pp(pascal_lower(5))
puts(1,"=== Pascal symmetrical matrix ===\n")
pp(pascal_symmetric(5))
