Pascal matrix generation

From Rosetta Code
Revision as of 01:08, 20 December 2021 by rosettacode>RobinVowels (→‎{{header|PL/I}}: New version added)
Task
Pascal matrix generation
You are encouraged to solve this task according to the task description, using any language you may know.

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.

11l

Translation of: Python

<lang 11l>F pascal_upp(n)

  V s = [[0] * n] * n
  s[0] = [1] * n
  L(i) 1 .< n
     L(j) i .< n
        s[i][j] = s[i - 1][j - 1] + s[i][j - 1]
  R s

F pascal_low(n)

  V upp = pascal_upp(n)
  V s = [[0] * n] * n
  L(x) 0 .< n
     L(y) 0 .< n
        s[y][x] = upp[x][y]
  R s

F pascal_sym(n)

  V s = [[1] * n] * n
  L(i) 1 .< n
     L(j) 1 .< n
        s[i][j] = s[i - 1][j] + s[i][j - 1]
  R s

F pp(mat)

  print(‘[’mat.map(String).join(",\n ")‘]’)

-V n = 5 print(‘Upper:’) pp(pascal_upp(n)) print("\nLower:") pp(pascal_low(n)) print("\nSymmetric:") pp(pascal_sym(n))</lang>

Output:
Upper:
[[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]]

Lower:
[[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]]

Symmetric:
[[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]]

360 Assembly

<lang 360asm>* Pascal matrix generation - 10/06/2018 PASCMATR CSECT

        USING  PASCMATR,R13       base register
        B      72(R15)            skip savearea
        DC     17F'0'             savearea
        SAVE   (14,12)            save previous context
        ST     R13,4(R15)         link backward
        ST     R15,8(R13)         link forward
        LR     R13,R15            set addressability
        MVC    MAT,=F'1'          mat(1,1)=1
        LA     R6,1               i=1
      DO WHILE=(C,R6,LE,N)        do i=1 to n;
        LA     R7,1                 j=1
      DO WHILE=(C,R7,LE,N)          do j=1 to n;       
        LR     R2,R6                  i
        LA     R3,1(R7)               r3=j+1
        LR     R1,R6                  i
        BCTR   R1,0                   -1
        MH     R1,NN                  *nn
        AR     R1,R7                  ~(i,j)
        SLA    R1,2                   *4
        L      R4,MAT-4(R1)           r4=mat(i,j)
        LR     R5,R6                  i
        MH     R5,NN                  *nn
        AR     R5,R7                  ~(i+1,j)
        SLA    R5,2                   *4
        L      R5,MAT-4(R5)           r5=mat(i+1,j)
        AR     R4,R5                  r4=mat(i,j)+mat(i+1,j)
        MH     R2,NN                  *nn
        AR     R2,R3                  ~(i+1,j+1)
        SLA    R2,2                   *4
        ST     R4,MAT-4(R2)           mat(i+1,j+1)=mat(i,j)+mat(i+1,j)
        LA     R7,1(R7)               j++
      ENDDO    ,                    enddo j
        LA     R6,1(R6)             i++
      ENDDO    ,                  enddo i
        MVC    TITLE,=CL20'Upper:'
        BAL    R14,PRINTMAT       call printmat
        MVC    MAT,=F'1'          mat(1,1)=1
        LA     R6,1               i=1
      DO WHILE=(C,R6,LE,N)        do i=1 to n;
        LA     R7,1                 j=1
      DO WHILE=(C,R7,LE,N)          do j=1 to n;       
        LR     R2,R6                  i
        LA     R3,1(R7)               r3=j+1
        LR     R1,R6                  i
        BCTR   R1,0                   -1
        MH     R1,NN                  *nn
        LR     R0,R7                  j
        AR     R1,R0                  ~(i,j)
        SLA    R1,2                   *4
        L      R4,MAT-4(R1)           r4=mat(i,j)
        LA     R5,1(R7)               j+1
        LR     R1,R6                  i
        BCTR   R1,0                   -1
        MH     R1,NN                  *nn
        AR     R1,R5                  ~(i,j+1)
        SLA    R1,2                   *4
        L      R5,MAT-4(R1)           r5=mat(i,j+1)
        AR     R4,R5                  mat(i,j)+mat(i,j+1)
        MH     R2,NN                  *nn
        AR     R2,R3                  ~(i+1,j+1)
        SLA    R2,2                   *4
        ST     R4,MAT-4(R2)           mat(i+1,j+1)=mat(i,j)+mat(i,j+1)
        LA     R7,1(R7)               j++
      ENDDO    ,                    enddo j
        LA     R6,1(R6)             i++
      ENDDO    ,                  enddo i
        MVC    TITLE,=CL20'Lower:'
        BAL    R14,PRINTMAT       call printmat
        MVC    MAT+24,=F'1'       mat(2,1)=1
        LA     R6,1               i=1
      DO WHILE=(C,R6,LE,N)        do i=1 to n;
        LA     R7,1                 j=1
      DO WHILE=(C,R7,LE,N)          do j=1 to n;       
        LR     R2,R6                  i
        LA     R3,1(R7)               r3=j+1                 j
        LR     R1,R6                  i
        BCTR   R1,0                   -1
        MH     R1,NN                  *nn
        AR     R1,R3                  ~(i,j+1)
        SLA    R1,2                   *4
        L      R4,MAT-4(R1)           r4=mat(i,j+1)
        LR     R5,R6                  i
        MH     R5,NN                  *nn
        AR     R5,R7                  j
        SLA    R5,2                   *4
        L      R5,MAT-4(R5)           r5=mat(i+1,j)
        AR     R4,R5                  mat(i,j+1)+mat(i+1,j)
        MH     R2,NN                  *nn
        AR     R2,R3                  ~(i+1,j+1)
        SLA    R2,2                   *4
        ST     R4,MAT-4(R2)         mat(i+1,j+1)=mat(i,j+1)+mat(i+1,j)
        LA     R7,1(R7)               j++
      ENDDO    ,                    enddo j
        LA     R6,1(R6)             i++
      ENDDO    ,                  enddo i
        MVC    TITLE,=CL20'Symmetric:'
        BAL    R14,PRINTMAT       call printmat
        L      R13,4(0,R13)       restore previous savearea pointer
        RETURN (14,12),RC=0       restore registers from calling sav

PRINTMAT XPRNT TITLE,L'TITLE print title -----------------------

        LA     R10,PG             pgi=0
        LA     R6,1               i=1
      DO WHILE=(C,R6,LE,N)        do i=1 to n;
        LA     R7,1                 j=1
      DO WHILE=(C,R7,LE,N)          do j=1 to n;       
        LR     R2,R6                  i
        LR     R3,R7                  j
        LA     R3,1(R3)               j+1
        MH     R2,NN                  *nn
        AR     R2,R3                  ~(i+1,j+1)
        SLA    R2,2                   *4
        L      R2,MAT-4(R2)           mat(i+1,j+1)
        XDECO  R2,XDEC                edit mat(i+1,j+1)
        MVC    0(5,R10),XDEC+7        output mat(i+1,j+1)
        LA     R10,5(R10)             pgi+=5
        LA     R7,1(R7)               j++
      ENDDO    ,                    enddo j
        XPRNT  PG,L'PG              print
        LA     R10,PG               pgi=0
        LA     R6,1(R6)             i++
      ENDDO    ,                  enddo i
        BR     R14                return to caller -------------------

X EQU 5 matrix size N DC A(X) n=x NN DC AL2(X+1) nn=x+1 MAT DC ((X+1)*(X+1))F'0' mat(x+1,x+1) TITLE DC CL20' ' title PG DC CL80' ' buffer PGI DC H'0' buffer index XDEC DS CL12 temp

        YREGS
        END    PASCMATR</lang>
Output:
Upper:
    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
Lower:
    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
Symmetric:
    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

Action!

<lang Action!>BYTE FUNC Index(BYTE i,j,dim) RETURN (i*dim+j)

PROC PascalUpper(BYTE ARRAY mat BYTE dim)

 BYTE i,j
 FOR i=0 TO dim-1
 DO
   FOR j=0 TO dim-1
   DO
     IF i>j THEN
       mat(Index(i,j,dim))=0
     ELSEIF i=j OR i=0 THEN
       mat(Index(i,j,dim))=1
     ELSE
       mat(Index(i,j,dim))=mat(Index(i-1,j-1,dim))+mat(Index(i,j-1,dim))
     FI
   OD
 OD

RETURN

PROC PascalLower(BYTE ARRAY mat BYTE dim)

 BYTE i,j
 FOR i=0 TO dim-1
 DO
   FOR j=0 TO dim-1
   DO
     IF i<j THEN
       mat(Index(i,j,dim))=0
     ELSEIF i=j OR j=0 THEN
       mat(Index(i,j,dim))=1
     ELSE
       mat(Index(i,j,dim))=mat(Index(i-1,j-1,dim))+mat(Index(i-1,j,dim))
     FI
   OD
 OD

RETURN

PROC PascalSymmetric(BYTE ARRAY mat BYTE dim)

 BYTE i,j
 FOR i=0 TO dim-1
 DO
   FOR j=0 TO dim-1
   DO
     IF i=0 OR j=0 THEN
       mat(Index(i,j,dim))=1
     ELSE
       mat(Index(i,j,dim))=mat(Index(i-1,j,dim))+mat(Index(i,j-1,dim))
     FI
   OD
 OD

RETURN

PROC PrintMatrix(BYTE ARRAY mat BYTE dim)

 BYTE i,j,v
 FOR i=0 TO dim-1
 DO
   FOR j=0 TO dim-1
   DO
     v=mat(Index(i,j,dim))
     IF v<10 THEN
       Print("   ")
     ELSEIF v<100 THEN
       Print("  ")
     FI
     PrintB(v)
   OD
   PutE()
 OD

RETURN

PROC Main()

 BYTE ARRAY mat(25)
 BYTE dim=[5]
 PrintE("Pascal upper matrix:")
 PascalUpper(mat,dim)
 PrintMatrix(mat,dim)
 PutE()
 PrintE("Pascal lower matrix:")
 PascalLower(mat,dim)
 PrintMatrix(mat,dim)
 PutE()
 PrintE("Pascal symmetric matrix:")
 PascalSymmetric(mat,dim)
 PrintMatrix(mat,dim)

RETURN</lang>

Output:

Screenshot from Atari 8-bit computer

Pascal upper matrix:
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

Pascal lower 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

Pascal symmetric matrix:
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

Ada

<lang ada>-- for I/O with Ada.Text_IO; use Ada.Text_IO; with Ada.Integer_Text_IO; use Ada.Integer_Text_IO; with Ada.Float_Text_IO; use Ada.Float_Text_IO;

-- for estimating the maximum width of a column with Ada.Numerics.Generic_Elementary_Functions;

procedure PascalMatrix is

 type Matrix is array (Positive range <>, Positive range <>) of Natural;
 -- instantiate Generic_Elementary_Functions for Float type
 package Math is new Ada.Numerics.Generic_Elementary_Functions(Float_Type => Float);
 use Math;
 
 procedure Print(m: in Matrix) is
   -- determine the maximum width of a column
   w: Float := Log(Float(m'Length(1)**(m'Length(1)/2)), 10.0);
   width: Positive := Natural(Float'Ceiling(w)) + 1;
   begin
     for i in m'First(1)..m'Last(1) loop
       Put("( ");
       for j in m'First(2)..m'Last(2) loop
         Put(m(i,j), width);
       end loop;
       Put(" )"); New_Line(1);
     end loop;
   end Print;
 
 function Upper_Triangular(n: in Positive) return Matrix is
   result: Matrix(1..n, 1..n) := (
                                   1 => ( others => 1 ),
                                   others => ( others => 0 )
                                 );
   begin
     for i in 2..n loop
       result(i,i) := 1;
       for j in i+1..n loop
         result(i,j) := result(i,j-1) + result(i-1,j-1);
       end loop;
     end loop;
     return result;
   end Upper_Triangular;
 
 function Lower_Triangular(n: in Positive) return Matrix is
   result: Matrix(1..n, 1..n) := (
                                   others => ( 1 => 1, others => 0 )
                                 );
   begin
     for i in 2..n loop
       result(i,i) := 1;
       for j in i+1..n loop
         result(j,i) := result(j-1,i) + result(j-1,i-1);
       end loop;
     end loop;
     return result;
   end Lower_Triangular;
 
 function Symmetric(n: in Positive) return Matrix is
   result: Matrix(1..n, 1..n) := (
                                  1 => ( others => 1 ),
                                  others => ( 1 => 1, others => 0 )
                                 );
   begin
     for i in 2..n loop
       for j in 2..n loop
         result(i,j) := result(i,j-1) + result(i-1,j);
       end loop;
     end loop;
     return result;
   end Symmetric;
 
 n: Positive;
 
 begin
   Put("What dimension Pascal matrix would you like? ");
   Get(n);
   Put("Upper triangular:"); New_Line(1);
   Print(Upper_Triangular(n));
   Put("Lower triangular:"); New_Line(1);
   Print(Lower_Triangular(n));
   Put("Symmetric:"); New_Line(1);
   Print(Symmetric(n));
 end PascalMatrix;</lang>
Output:
What dimension Pascal matrix would you like? 5
Upper triangular:
(   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 )
Lower triangular:
(   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 )
Symmetric:
(   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 )

ALGOL 68

<lang algol68>BEGIN

   # returns an upper Pascal matrix of size n #
   PROC upper pascal matrix = ( INT n )[,]INT:
        BEGIN
           [ 1 : n, 1 : n ]INT result;
           FOR j        TO n DO result[ 1, j ] := 1 OD;
           FOR i FROM 2 TO n DO
               result[ i, 1 ] := 0;
               FOR j FROM 2 TO n DO
                   result[ i, j ] := result[ i - 1, j - 1 ] + result[ i, j - 1 ]
               OD
           OD;
           result
        END # upper pascal matrix # ;
   # returns a lower Pascal matrix of size n #
   PROC lower pascal matrix = ( INT n )[,]INT:
        BEGIN
           [ 1 : n, 1 : n ]INT result;
           FOR i        TO n DO result[ i, 1 ] := 1 OD;
           FOR j FROM 2 TO n DO
               result[ 1, j ] := 0;
               FOR i FROM 2 TO n DO
                   result[ i, j ] := result[ i - 1, j - 1 ] + result[ i - 1, j ]
               OD
           OD;
           result
        END # lower pascal matrix # ;
   # returns a symmetric Pascal matrix of size n #
   PROC symmetric pascal matrix = ( INT n )[,]INT:
        BEGIN
           [ 1 : n, 1 : n ]INT result;
           FOR i TO n DO
               result[ i, 1 ] := 1;
               result[ 1, i ] := 1
           OD;
           FOR j FROM 2 TO n DO
               FOR i FROM 2 TO n DO
                   result[ i, j ] := result[ i, j - 1 ] + result[ i - 1, j ]
               OD
           OD;
           result
        END # symmetric pascal matrix # ;
   # print the matrix m with the specified field width #
   PROC print matrix = ( [,]INT m, INT field width )VOID:
        BEGIN
            FOR i FROM 1 LWB m TO 1 UPB m DO
                FOR j FROM 2 LWB m TO 2 UPB m DO
                    print( ( " ", whole( m[ i, j ], - field width ) ) )
                OD;
                print( ( newline ) )
            OD
        END # print matrix # ;
   print( ( "upper:",     newline ) ); print matrix( upper pascal matrix(     5 ), 2 );
   print( ( "lower:",     newline ) ); print matrix( lower pascal matrix(     5 ), 2 );
   print( ( "symmetric:", newline ) ); print matrix( symmetric pascal matrix( 5 ), 2 )

END</lang>

Output:
upper:
  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
lower:
  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
symmetric:
  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

ALGOL W

Translation of: ALGOL_68

<lang algolw>begin

   % initialises m to an upper Pascal matrix of size n %
   % the bounds of m must be at least 1 :: n, 1 :: n   %
   procedure upperPascalMatrix ( integer array m( *, * )
                               ; integer value n
                               ) ;
   begin
       for j := 1 until n do m( 1, j ) := 1;
       for i := 2 until n do begin
           m( i, 1 ) := 0;
           for j := 2 until n do m( i, j ) := m( i - 1, j - 1 ) + m( i, j - 1 )
       end for_i
   end upperPascalMatrix ;
   % initialises m to a lower Pascal matrix of size n  %
   % the bounds of m must be at least 1 :: n, 1 :: n   %
   procedure lowerPascalMatrix ( integer array m( *, * )
                              ; integer value n
                              ) ;
   begin
       for i := 1 until n do m( i, 1 ) := 1;
       for j := 2 until n do begin
           m( 1, j ) := 0;
           for i := 2 until n do m( i, j ) := m( i - 1, j - 1 ) + m( i - 1, j )
       end for_j
   end lowerPascalMatrix ;
   % initialises m to a symmetric Pascal matrix of size n %
   % the bounds of m must be at least 1 :: n, 1 :: n   %
   procedure symmetricPascalMatrix ( integer array m( *, * )
                                   ; integer value n
                                   ) ;
   begin
       for i := 1 until n do begin
           m( i, 1 ) := 1;
           m( 1, i ) := 1
       end for_i;
       for j := 2 until n do for i := 2 until n do m( i, j ) := m( i, j - 1 ) + m( i - 1, j )
   end symmetricPascalMatrix ;
   begin % test the pascal matrix procedures %
       % print the matrix m with the specified field width %
       % the bounds of m must be at least 1 :: n, 1 :: n   %
       procedure printMatrix ( integer array m( *, * )
                             ; integer value n
                             ; integer value fieldWidth
                             ) ;
       begin
           for i := 1 until n do begin
               write(                         i_w := fieldWidth, s_w := 0, " ", m( i, 1 ) );
               for j := 2 until n do writeon( i_w := fieldWidth, s_w := 0, " ", m( i, j ) )
           end for_i
       end printMatrix ;
       integer array m( 1 :: 10, 1 :: 10 );
       integer n, w;
       n := 5; w := 2;
       upperPascalMatrix(     m, n ); write( "upper:"     ); printMatrix( m, n, w );
       lowerPascalMatrix(     m, n ); write( "lower:"     ); printMatrix( m, n, w );
       symmetricPascalMatrix( m, n ); write( "symmetric:" ); printMatrix( m, n, w )
   end

end.</lang>

Output:
upper:
  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
lower:
  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
symmetric:
  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

APL

Works with: Dyalog APL

<lang APL>upper ← ∘.!⍨¯1+⍳ lower ← ⍉(∘.!⍨¯1+⍳) symmetric ← (⊢![1]∘.+⍨)¯1+⍳</lang>

Output:
      ((⊂upper),(⊂lower),(⊂symmetric)) 5
┌─────────┬─────────┬────────────┐
│1 1 1 1 1│1 0 0 0 0│1 1  1  1  1│
│0 1 2 3 4│1 1 0 0 0│1 2  3  4  5│
│0 0 1 3 6│1 2 1 0 0│1 3  6 10 15│
│0 0 0 1 4│1 3 3 1 0│1 4 10 20 35│
│0 0 0 0 1│1 4 6 4 1│1 5 15 35 70│
└─────────┴─────────┴────────────┘

AppleScript

By composition of generic functions: <lang AppleScript>-- PASCAL MATRIX -------------------------------------------------------------

-- pascalMatrix :: ((Int, Int) -> (Int, Int)) -> Int -> Int on pascalMatrix(f, n)

   chunksOf(n, map(compose(my bc, f), range({{0, 0}, {n - 1, n - 1}})))

end pascalMatrix

-- Binomial coefficient -- bc :: (Int, Int) -> Int on bc(nk)

   set {n, k} to nk
   script bc_
       on |λ|(a, x)
           floor((a * (n - x + 1)) / x)
       end |λ|
   end script
   foldl(bc_, 1, enumFromTo(1, k))

end bc


-- TEST ---------------------------------------------------------------------- on run

   set matrixSize to 5
   
   script symm
       on |λ|(ab)
           set {a, b} to ab
           {a + b, a}
       end |λ|
   end script
   
   script format
       on |λ|(s, xs)
           unlines(concat({{s}, map(my show, xs), {""}}))
       end |λ|
   end script
   
   unlines(zipWith(format, ¬
       {"Lower", "Upper", "Symmetric"}, ¬
       |<*>|(map(curry(pascalMatrix), [|id|, swap, symm]), {matrixSize})))

end run


-- GENERIC FUNCTIONS ---------------------------------------------------------

-- A list of functions applied to a list of arguments -- (<*> | ap) :: [(a -> b)] -> [a] -> [b] on |<*>|(fs, xs)

   set {nf, nx} to {length of fs, length of xs}
   set acc to {}
   repeat with i from 1 to nf
       tell mReturn(item i of fs)
           repeat with j from 1 to nx
               set end of acc to |λ|(contents of (item j of xs))
           end repeat
       end tell
   end repeat
   return acc

end |<*>|

-- chunksOf :: Int -> [a] -> a on chunksOf(k, xs)

   script
       on go(ys)
           set {a, b} to splitAt(k, ys)
           if isNull(a) then
               {}
           else
               {a} & go(b)
           end if
       end go
   end script
   result's go(xs)

end chunksOf

-- compose :: (b -> c) -> (a -> b) -> (a -> c) on compose(f, g)

   script
       on |λ|(x)
           mReturn(f)'s |λ|(mReturn(g)'s |λ|(x))
       end |λ|
   end script

end compose

-- concat :: a -> [a] | [String] -> String on concat(xs)

   if length of xs > 0 and class of (item 1 of xs) is string then
       set acc to ""
   else
       set acc to {}
   end if
   repeat with i from 1 to length of xs
       set acc to acc & item i of xs
   end repeat
   acc

end concat

-- cons :: a -> [a] -> [a] on cons(x, xs)

   {x} & xs

end cons

-- curry :: (Script|Handler) -> Script on curry(f)

   script
       on |λ|(a)
           script
               on |λ|(b)
                   |λ|(a, b) of mReturn(f)
               end |λ|
           end script
       end |λ|
   end script

end curry

-- enumFromTo :: Int -> Int -> [Int] on enumFromTo(m, n)

   set lst to {}
   repeat with i from m to n
       set end of lst to i
   end repeat
   return lst

end enumFromTo

-- floor :: Num -> Int on floor(x)

   if x < 0 and x mod 1 is not 0 then
       (x div 1) - 1
   else
       (x div 1)
   end if

end floor

-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs)

   tell mReturn(f)
       set v to startValue
       set lng to length of xs
       repeat with i from 1 to lng
           set v to |λ|(v, item i of xs, i, xs)
       end repeat
       return v
   end tell

end foldl

-- foldr :: (b -> a -> a) -> a -> [b] -> a on foldr(f, startValue, xs)

   tell mReturn(f)
       set v to startValue
       set lng to length of xs
       repeat with i from lng to 1 by -1
           set v to |λ|(item i of xs, v, i, xs)
       end repeat
       return v
   end tell

end foldr

-- id :: a -> a on |id|(x)

   x

end |id|

-- intercalate :: Text -> [Text] -> Text on intercalate(strText, lstText)

   set {dlm, my text item delimiters} to {my text item delimiters, strText}
   set strJoined to lstText as text
   set my text item delimiters to dlm
   return strJoined

end intercalate

-- isNull :: [a] -> Bool on isNull(xs)

   if class of xs is string then
       xs = ""
   else
       xs = {}
   end if

end isNull

-- map :: (a -> b) -> [a] -> [b] on map(f, xs)

   tell mReturn(f)
       set lng to length of xs
       set lst to {}
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs, i, xs)
       end repeat
       return lst
   end tell

end map

-- min :: Ord a => a -> a -> a on min(x, y)

   if y < x then
       y
   else
       x
   end if

end min

-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)

   if class of f is script then
       f
   else
       script
           property |λ| : f
       end script
   end if

end mReturn

-- quot :: Int -> Int -> Int on quot(m, n)

   m div n

end quot

-- range :: Ix a => (a, a) -> [a] on range({a, b})

   if class of a is list then
       set {xs, ys} to {a, b}
   else
       set {xs, ys} to {{a}, {b}}
   end if
   set lng to length of xs
   
   if lng = length of ys then
       if lng > 1 then
           script
               on |λ|(_, i)
                   enumFromTo(item i of xs, item i of ys)
               end |λ|
           end script
           sequence(map(result, xs))
       else
           enumFromTo(a, b)
       end if
   else
       {}
   end if

end range

-- sequence :: Monad m => [m a] -> m [a] -- sequence :: [a] -> a on sequence(xs)

   traverse(|id|, xs)

end sequence

-- show :: a -> String on show(e)

   set c to class of e
   if c = list then
       script serialized
           on |λ|(v)
               show(v)
           end |λ|
       end script
       
       "[" & intercalate(", ", map(serialized, e)) & "]"
   else if c = record then
       script showField
           on |λ|(kv)
               set {k, ev} to kv
               "\"" & k & "\":" & show(ev)
           end |λ|
       end script
       
       "{" & intercalate(", ", ¬
           map(showField, zip(allKeys(e), allValues(e)))) & "}"
   else if c = date then
       "\"" & iso8601Z(e) & "\""
   else if c = text then
       "\"" & e & "\""
   else if (c = integer or c = real) then
       e as text
   else if c = class then
       "null"
   else
       try
           e as text
       on error
           ("«" & c as text) & "»"
       end try
   end if

end show

-- splitAt :: Int -> [a] -> ([a],[a]) on splitAt(n, xs)

   if n > 0 and n < length of xs then
       if class of xs is text then
           {items 1 thru n of xs as text, items (n + 1) thru -1 of xs as text}
       else
           {items 1 thru n of xs, items (n + 1) thru -1 of xs}
       end if
   else
       if n < 1 then
           {{}, xs}
       else
           {xs, {}}
       end if
   end if

end splitAt

-- swap :: (a, b) -> (b, a) on swap(ab)

   set {a, b} to ab
   {b, a}

end swap

-- traverse :: (a -> [b]) -> [a] -> b on traverse(f, xs)

   script
       property mf : mReturn(f)
       on |λ|(x, a)
           |<*>|(map(curry(cons), mf's |λ|(x)), a)
       end |λ|
   end script
   foldr(result, {{}}, xs)

end traverse

-- unlines :: [String] -> String on unlines(xs)

   intercalate(linefeed, xs)

end unlines

-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] on zipWith(f, xs, ys)

   set lng to min(length of xs, length of ys)
   set lst to {}
   tell mReturn(f)
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs, item i of ys)
       end repeat
       return lst
   end tell

end zipWith</lang>

Output:
Lower
[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]

Upper
[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]

Symmetric
[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]

AutoHotkey

<lang AutoHotkey>n := 5 MsgBox, 262144, ,% "" . "Pascal upper-triangular :`n" show(Pascal_Upper(n)) . "`n`nPascal lower-triangular :`n" show(Pascal_Lower(n)) . "`n`nPascal symmetric:`n" show(Pascal_Symm(n)) return

show(obj){ for i, o in obj{ line := "" for j, v in o line .= v ", " res .= "[" Trim(line, ", ") "]`n," } return "[" Trim(res, "`n,") "]" }

Pascal_Upper(n){ obj := fillObj(n) loop % n obj[1, A_Index] := 1 loop % n-1 obj[A_Index+1, 1] := 0 for i, o in obj for j, v in o if !(i = 1 or j = 1) obj[i, j] := obj[i, j-1] + obj[i-1, j-1] return obj }

Pascal_Lower(n){ obj := fillObj(n) loop % n obj[A_Index, 1] := 1 loop % n-1 obj[1, A_Index+1] := 0 for i, o in obj for j, v in o if !(i = 1 or j = 1) obj[i, j] := obj[i-1, j] + obj[i-1, j-1] return obj }

Pascal_Symm(n){ obj := fillObj(n) loop % n obj[A_Index, 1] := 1 loop % n-1 obj[1, A_Index+1] := 1 for i, o in obj for j, v in o if !(i = 1 or j = 1) obj[i, j] := obj[i-1, j] + obj[i, j-1] return obj }

fillObj(n){ obj := [] loop % n{ i := A_Index loop % n obj[i, A_Index] := 0 } return obj }</lang>

Output:
Pascal upper-triangular :
[[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]]

Pascal lower-triangular :
[[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]]

Pascal symmetric:
[[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]]

BASIC

<lang basic>10 DEFINT A-Z: S=5: DIM M(S,S) 20 PRINT "Lower-triangular matrix:": GOSUB 200: GOSUB 100 30 PRINT "Upper-triangular matrix:": GOSUB 300: GOSUB 100 40 PRINT "Symmetric matrix:": GOSUB 400: GOSUB 100 50 END 100 REM *** Print the matrix M *** 110 FOR Y=1 TO S 120 FOR X=1 TO S 130 PRINT USING " ##";M(X,Y); 140 NEXT X 150 PRINT 160 NEXT Y 170 PRINT 180 RETURN 200 REM *** Generate the lower-triangular matrix *** 210 FOR X=1 TO S: FOR Y=1 TO S 220 ON -(X>Y)-2*(X=Y OR X=1) GOTO 240,250 230 M(X,Y)=M(X-1,Y-1)+M(X,Y-1): GOTO 260 240 M(X,Y)=0: GOTO 260 250 M(X,Y)=1: GOTO 260 260 NEXT Y,X 270 RETURN 300 REM *** Generate the upper-triangular matrix *** 310 FOR X=1 TO S: FOR Y=1 TO S 320 ON -(X<Y)-2*(X=Y OR Y=1) GOTO 340,350 330 M(X,Y)=M(X-1,Y-1)+M(X-1,Y): GOTO 360 340 M(X,Y)=0: GOTO 360 350 M(X,Y)=1: GOTO 360 360 NEXT Y,X 370 RETURN 400 REM *** Generate the symmetric matrix *** 410 FOR X=1 TO S: FOR Y=1 TO S 420 IF X=1 OR Y=1 THEN M(X,Y)=1 ELSE M(X,Y)=M(X-1,Y)+M(X,Y-1) 430 NEXT Y,X 440 RETURN</lang>

Output:
Lower-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

Upper-triangular matrix:
  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

Symmetric matrix:
  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

BCPL

<lang bcpl>get "libhdr" manifest $( size = 5 $)

// Matrix index let ix(mat, n, x, y) = mat+y*n+x

let lower(m, n) be

   for y=0 to n-1
       for x=0 to n-1 do
           !ix(m,n,x,y) :=
               x>y       -> 0,
               x=y | x=0 -> 1,
               !ix(m,n,x-1,y-1) + !ix(m,n,x,y-1) 
               
            

let upper(m, n) be

   for y=0 to n-1
       for x=0 to n-1 do
           !ix(m,n,x,y) :=
               x<y       -> 0,
               x=y | y=0 -> 1,
               !ix(m,n,x-1,y-1) + !ix(m,n,x-1,y)
      

let symmetric(m, n) be

   for y=0 to n-1
       for x=0 to n-1 do
           !ix(m,n,x,y) :=
               x=0 | y=0 -> 1,
               !ix(m,n,x-1,y) + !ix(m,n,x,y-1)

// Print matrix let writemat(m, n, d) be

   for y=0 to n-1
   $(  for x=0 to n-1
       $(  writed(!ix(m,n,x,y), d)
           wrch(' ')
       $)
       wrch('*N')
   $)
   

// Generate and print 5-by-5 matrices let start() be $( let mat = vec size * size

   writes("Upper-triangular matrix:*N")
   upper(mat, size) ; writemat(mat, size, 2)
   
   writes("*NLower-triangular matrix:*N")
   lower(mat, size) ; writemat(mat, size, 2)
   
   writes("*NSymmetric matrix:*N")
   symmetric(mat, size) ; writemat(mat, size, 2)

$)</lang>

Output:
Upper-triangular matrix:
 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

Lower-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

Symmetric matrix:
 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

C

<lang c>

  1. include <stdio.h>
  2. include <stdlib.h>

void pascal_low(int **mat, int n) {

   int i, j;
   for (i = 0; i < n; ++i)
       for (j = 0; j < n; ++j)
           if (i < j)
               mat[i][j] = 0;
           else if (i == j || j == 0)
               mat[i][j] = 1;
           else
               mat[i][j] = mat[i - 1][j - 1] + mat[i - 1][j];

}

void pascal_upp(int **mat, int n) {

   int i, j;
   for (i = 0; i < n; ++i)
       for (j = 0; j < n; ++j)
           if (i > j)
               mat[i][j] = 0;
           else if (i == j || i == 0)
               mat[i][j] = 1;
           else
               mat[i][j] = mat[i - 1][j - 1] + mat[i][j - 1];

}

void pascal_sym(int **mat, int n) {

   int i, j;
   for (i = 0; i < n; ++i)
       for (j = 0; j < n; ++j)
           if (i == 0 || j == 0)
               mat[i][j] = 1;
           else
               mat[i][j] = mat[i - 1][j] + mat[i][j - 1];

}

int main(int argc, char * argv[]) {

   int **mat;
   int i, j, n;
   /* Input size of the matrix */
   n = 5;
   /* Matrix allocation */
   mat = calloc(n, sizeof(int *));
   for (i = 0; i < n; ++i)
       mat[i] = calloc(n, sizeof(int));
   /* Matrix computation */
   printf("=== Pascal upper matrix ===\n");
   pascal_upp(mat, n);
   for (i = 0; i < n; i++)
       for (j = 0; j < n; j++)
           printf("%4d%c", mat[i][j], j < n - 1 ? ' ' : '\n');
   printf("=== Pascal lower matrix ===\n");
   pascal_low(mat, n);
   for (i = 0; i < n; i++)
       for (j = 0; j < n; j++)
           printf("%4d%c", mat[i][j], j < n - 1 ? ' ' : '\n');
   printf("=== Pascal symmetric matrix ===\n");
   pascal_sym(mat, n);
   for (i = 0; i < n; i++)
       for (j = 0; j < n; j++)
           printf("%4d%c", mat[i][j], j < n - 1 ? ' ' : '\n');
   return 0;

} </lang>

Output:
=== Pascal upper matrix ===
   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
=== Pascal lower 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
=== Pascal symmetric matrix ===
   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

C#

<lang csharp>using System;

public static class PascalMatrixGeneration {

   public static void Main() {
       Print(GenerateUpper(5));
       Console.WriteLine();
       Print(GenerateLower(5));
       Console.WriteLine();
       Print(GenerateSymmetric(5));
   }
   static int[,] GenerateUpper(int size) {
       int[,] m = new int[size, size];
       for (int c = 0; c < size; c++) m[0, c] = 1;
       for (int r = 1; r < size; r++) {
           for (int c = r; c < size; c++) {
               m[r, c] = m[r-1, c-1] + m[r, c-1];
           }
       }
       return m;
   }
   static int[,] GenerateLower(int size) {
       int[,] m = new int[size, size];
       for (int r = 0; r < size; r++) m[r, 0] = 1;
       for (int c = 1; c < size; c++) {
           for (int r = c; r < size; r++) {
               m[r, c] = m[r-1, c-1] + m[r-1, c];
           }
       }
       return m;
   }
   static int[,] GenerateSymmetric(int size) {
       int[,] m = new int[size, size];
       for (int i = 0; i < size; i++) m[0, i] = m[i, 0] = 1;
       for (int r = 1; r < size; r++) {
           for (int c = 1; c < size; c++) {
               m[r, c] = m[r-1, c] + m[r, c-1];
           }
       }
       return m;
   }
   static void Print(int[,] matrix) {
       string[,] m = ToString(matrix);
       int width = m.Cast<string>().Select(s => s.Length).Max();
       int rows = matrix.GetLength(0), columns = matrix.GetLength(1);
       for (int row = 0; row < rows; row++) {
           Console.WriteLine("|" + string.Join(" ", Range(0, columns).Select(column => m[row, column].PadLeft(width, ' '))) + "|");
       }
   }
   static string[,] ToString(int[,] matrix) {
       int rows = matrix.GetLength(0), columns = matrix.GetLength(1);
       string[,] m = new string[rows, columns];
       for (int r = 0; r < rows; r++) {
           for (int c = 0; c < columns; c++) {
               m[r, c] = matrix[r, c].ToString();
           }
       }
       return m;
   }
   

}</lang>

Output:
|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|

|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|

| 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|

C++

Works with: GCC version version 7.2.0 (Ubuntu 7.2.0-8ubuntu3.2)

<lang cpp>#include <iostream>

  1. include <vector>

typedef std::vector<std::vector<int>> vv;

vv pascal_upper(int n) {

   vv matrix(n);
   for (int i = 0; i < n; ++i) {
       for (int j = 0; j < n; ++j) {
               if (i > j) matrix[i].push_back(0);
               else if (i == j || i == 0) matrix[i].push_back(1);
               else matrix[i].push_back(matrix[i - 1][j - 1] + matrix[i][j - 1]);
           }
       }
       return matrix;
   }

vv pascal_lower(int n) {

   vv matrix(n);
   for (int i = 0; i < n; ++i) {
       for (int j = 0; j < n; ++j) {
           if (i < j) matrix[i].push_back(0);
           else if (i == j || j == 0) matrix[i].push_back(1);
           else matrix[i].push_back(matrix[i - 1][j - 1] + matrix[i - 1][j]);
       }
   }
   return matrix;

}

vv pascal_symmetric(int n) {

   vv matrix(n);
   for (int i = 0; i < n; ++i) {
       for (int j = 0; j < n; ++j) {
           if (i == 0 || j == 0) matrix[i].push_back(1);
           else matrix[i].push_back(matrix[i][j - 1] + matrix[i - 1][j]);
       }
   }
   return matrix;

}


void print_matrix(vv matrix) {

   for (std::vector<int> v: matrix) {
       for (int i: v) {
           std::cout << " " << i;
       }
       std::cout << std::endl;
   }

}

int main() {

   std::cout << "PASCAL UPPER MATRIX" << std::endl;
   print_matrix(pascal_upper(5));
   std::cout << "PASCAL LOWER MATRIX" << std::endl;
   print_matrix(pascal_lower(5));
   std::cout << "PASCAL SYMMETRIC MATRIX" << std::endl;
   print_matrix(pascal_symmetric(5));

}</lang>

Output:
PASCAL UPPER MATRIX
 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
PASCAL LOWER 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
PASCAL SYMMETRIC MATRIX
 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

Clojure

<lang clojure>(defn binomial-coeff [n k]

 (reduce #(quot (* %1 (inc (- n %2))) %2)
         1
         (range 1 (inc k))))

(defn pascal-upper [n]

 (map
  (fn [i]
    (map (fn [j]
           (binomial-coeff j i))
         (range n)))
  (range n)))

(defn pascal-lower [n]

 (map
  (fn [i]
    (map (fn [j]
           (binomial-coeff i j))
         (range n)))
  (range n)))

(defn pascal-symmetric [n]

 (map
  (fn [i]
    (map (fn [j]
           (binomial-coeff (+ i j) i))
         (range n)))
  (range n)))

(defn pascal-matrix [n]

 (println "Upper:")
 (run! println (pascal-upper n))
 (println)
 (println "Lower:")
 (run! println (pascal-lower n))
 (println)
 (println "Symmetric:")
 (run! println (pascal-symmetric n)))</lang>
Output:
=> (pascal-matrix 5)
Upper:
(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)

Lower:
(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)

Symmetric:
(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)

CLU

<lang clu>matrix = array[array[int]]

make_matrix = proc (gen: proctype (int,int,matrix) returns (int),

                   size: int) returns (matrix)
   m: matrix := matrix$fill_copy(0, size, array[int]$fill(0, size, 0))
   for y: int in int$from_to(0, size-1) do
       for x: int in int$from_to(0, size-1) do
           m[y][x] := gen(x,y,m)
       end
   end
   return(m)

end make_matrix

lower = proc (x,y: int, m: matrix) returns (int)

   if x>y then return(0)
   elseif x=y | x=0 then return(1)
   else return( m[y-1][x-1] + m[y-1][x] )
   end

end lower

upper = proc (x,y: int, m: matrix) returns (int)

   if x<y then return(0)
   elseif x=y | y=0 then return(1)
   else return( m[y-1][x-1] + m[y][x-1] )
   end

end upper

symmetric = proc (x,y: int, m: matrix) returns (int)

   if x=0 | y=0 then return(1)
   else return(m[y][x-1] + m[y-1][x])
   end

end symmetric

print_matrix = proc (s: stream, m: matrix, w: int)

   for line: array[int] in matrix$elements(m) do
       for item: int in array[int]$elements(line) do
           stream$putright(s, int$unparse(item), w)
           stream$putc(s, ' ')
       end
       stream$putl(s, "")
   end

end print_matrix

start_up = proc ()

   po: stream := stream$primary_output()
   
   stream$putl(po, "Upper-triangular matrix:")
   print_matrix(po, make_matrix(upper,5), 1)
   
   stream$putl(po, "\nLower-triangular matrix:")
   print_matrix(po, make_matrix(lower,5), 1)
   
   stream$putl(po, "\nSymmetric matrix:")
   print_matrix(po, make_matrix(symmetric,5), 2)

end start_up</lang>

Output:
Upper-triangular matrix:
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

Lower-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

Symmetric matrix:
 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

Common Lisp

<lang lisp>(defun pascal-lower (n &aux (a (make-array (list n n) :initial-element 0)))

   (dotimes (i n)
       (setf (aref a i 0) 1))
   (dotimes (i (1- n) a)
       (dotimes (j (1- n))
           (setf (aref a (1+ i) (1+ j))
               (+ (aref a i j)
                  (aref a i (1+ j)))))))
                  

(defun pascal-upper (n &aux (a (make-array (list n n) :initial-element 0)))

   (dotimes (i n)
       (setf (aref a 0 i) 1))
   (dotimes (i (1- n) a)
       (dotimes (j (1- n))
           (setf (aref a (1+ j) (1+ i))
               (+ (aref a j i)
                  (aref a (1+ j) i))))))

(defun pascal-symmetric (n &aux (a (make-array (list n n) :initial-element 0)))

   (dotimes (i n)
       (setf (aref a i 0) 1 (aref a 0 i) 1))
   (dotimes (i (1- n) a)
       (dotimes (j (1- n))
           (setf (aref a (1+ i) (1+ j))
               (+ (aref a (1+ i) j)
                  (aref a i (1+ j)))))))

? (pascal-lower 4)

  1. 2A((1 0 0 0) (1 1 0 0) (1 2 1 0) (1 3 3 1))

? (pascal-upper 4)

  1. 2A((1 1 1 1) (0 1 2 3) (0 0 1 3) (0 0 0 1))

? (pascal-symmetric 4)

  1. 2A((1 1 1 1) (1 2 3 4) (1 3 6 10) (1 4 10 20))
In case one really insists in printing the array row by row

(defun print-matrix (a)

   (let ((p (array-dimension a 0))
         (q (array-dimension a 1)))
       (dotimes (i p)
           (dotimes (j q)
               (princ (aref a i j))
               (princ #\Space))
           (terpri))))

? (print-matrix (pascal-lower 5)) 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

? (print-matrix (pascal-upper 5)) 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

? (print-matrix (pascal-symmetric 5)) 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</lang>

D

Translation of: Python

<lang d>import std.stdio, std.bigint, std.range, std.algorithm;

auto binomialCoeff(in uint n, in uint k) pure nothrow {

   BigInt result = 1;
   foreach (immutable i; 1 .. k + 1)
       result = result * (n - i + 1) / i;
   return result;

}

auto pascalUpp(in uint n) pure nothrow {

   return n.iota.map!(i => n.iota.map!(j => binomialCoeff(j, i)));

}

auto pascalLow(in uint n) pure nothrow {

   return n.iota.map!(i => n.iota.map!(j => binomialCoeff(i, j)));

}

auto pascalSym(in uint n) pure nothrow {

   return n.iota.map!(i => n.iota.map!(j => binomialCoeff(i + j, i)));

}

void main() {

   enum n = 5;
   writefln("Upper:\n%(%(%2d %)\n%)", pascalUpp(n));
   writefln("\nLower:\n%(%(%2d %)\n%)", pascalLow(n));
   writefln("\nSymmetric:\n%(%(%2d %)\n%)", pascalSym(n));

}</lang>

Output:
Upper:
 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

Lower:
 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

Symmetric:
 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

Delphi

See Pascal.

Elixir

<lang elixir>defmodule Pascal do

 defp ij(n), do: for i <- 1..n, j <- 1..n, do: {i,j}
 
 def upper_triangle(n) do
   Enum.reduce(ij(n), Map.new, fn {i,j},acc ->
     val = cond do
             i==1 -> 1
             j 0
             true -> Map.get(acc, {i-1, j-1}) + Map.get(acc, {i, j-1})
           end
     Map.put(acc, {i,j}, val)
   end) |> print(1..n)
 end
 
 def lower_triangle(n) do
   Enum.reduce(ij(n), Map.new, fn {i,j},acc ->
     val = cond do
             j==1 -> 1
             i<j  -> 0
             true -> Map.get(acc, {i-1, j-1}) + Map.get(acc, {i-1, j})
           end
     Map.put(acc, {i,j}, val)
   end) |> print(1..n)
 end
 
 def symmetic_triangle(n) do
   Enum.reduce(ij(n), Map.new, fn {i,j},acc ->
     val = if i==1 or j==1, do: 1,
                          else: Map.get(acc, {i-1, j}) + Map.get(acc, {i, j-1})
     Map.put(acc, {i,j}, val)
   end) |> print(1..n)
 end
 
 def print(matrix, range) do
   Enum.each(range, fn i ->
     Enum.map(range, fn j -> Map.get(matrix, {i,j}) end) |> IO.inspect
   end)
 end

end

IO.puts "Pascal upper-triangular matrix:" Pascal.upper_triangle(5) IO.puts "Pascal lower-triangular matrix:" Pascal.lower_triangle(5) IO.puts "Pascal symmetric matrix:" Pascal.symmetic_triangle(5)</lang>

Output:
Pascal upper-triangular matrix:
[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]
Pascal lower-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]
Pascal symmetric matrix:
[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]

Excel

LAMBDA

Binding the names PASCALMATRIX, BINCOEFF and SYMMETRIC to the following lambda expressions in the Name Manager of the Excel WorkBook:

(See LAMBDA: The ultimate Excel worksheet function)

<lang lisp>PASCALMATRIX =LAMBDA(n,

   LAMBDA(f,
       LET(
           ixs, SEQUENCE(n, n, 0, 1),
           f(BINCOEFF)(
               QUOTIENT(ixs, n)
           )(
               MOD(ixs, n)
           )
       )
   )

)


BINCOEFF =LAMBDA(n,

   LAMBDA(k,
       IF(n < k,
           0,
           QUOTIENT(FACT(n), FACT(k) * FACT(n - k))
       )
   )

)


SYMMETRIC =LAMBDA(f,

   LAMBDA(a,
       LAMBDA(b,
           f(a + b)(b)
       )
   )

)</lang>

and also assuming the following generic bindings in the Name Manager for the WorkBook:

<lang lisp>FLIP =LAMBDA(f,

   LAMBDA(a,
       LAMBDA(b,
           f(b)(a)
       )
   )

)


ID =LAMBDA(x, x)</lang>

Output:
fx =PASCALMATRIX(5)( ID )
A B C D E F G H
1 Lower
2 1 0 0 0 0
3 1 1 0 0 0
4 1 2 1 0 0
5 1 3 3 1 0
6 1 4 6 4 1
fx =PASCALMATRIX(5)( FLIP )
A B C D E F G H
1 Upper
2 1 1 1 1 1
3 0 1 2 3 4
4 0 0 1 3 6
5 0 0 0 1 4
6 0 0 0 0 1
fx =PASCALMATRIX(5)( SYMMETRIC )
A B C D E F G H
1 Symmetric
2 1 1 1 1 1
3 1 2 3 4 5
4 1 3 6 10 15
5 1 4 10 20 35
6 1 5 15 35 70

Factor

Works with: Factor version 0.99 2020-01-23

<lang factor>USING: arrays fry io kernel math math.combinatorics math.matrices prettyprint sequences ;

pascal ( n quot -- m )
   [ dup 2array <coordinate-matrix> ] dip
   '[ first2 @ nCk ] matrix-map ; inline
lower ( n -- m ) [ ] pascal ;
upper ( n -- m ) lower flip ;
symmetric ( n -- m ) [ [ + ] keep ] pascal ;

5 [ lower "Lower:" ] [ upper "Upper:" ] [ symmetric "Symmetric:" ] tri [ print simple-table. nl ] 2tri@</lang>

Output:
Lower:
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

Upper:
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

Symmetric:
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

Fermat

<lang fermat>&a; {set mode to 0-indexed matrices} Func Pasmat( n, t ) =

   ;{create a Pascal matrix of size n by n}
   ;{t=0 -> upper triangular, 1 -> lower triangular,2->symmetric}
   Array m[n, n];    {result is stored in array m}
   if t = 0 then
       [m]:=[<i=0,n-1><j=0,n-1> Bin(j,i) ];
   fi;
   if t = 1 then
       [m]:=[<i=0,n-1><j=0,n-1> Bin(i,j) ];
   fi;
   if t = 2 then
       [m]:=[<i=0,n-1><j=0,n-1> Bin(i+j,i) ];
   fi;

.;

Pasmat(5, 0); !!([m); !; Pasmat(5, 1); !!([m); !; Pasmat(5, 2); !!([m);</lang>

Output:

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

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

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

Fortran

The following program uses features of Fortran 2003.

<lang fortran>module pascal

implicit none

contains

   function pascal_lower(n) result(a)
       integer :: n, i, j
       integer, allocatable :: a(:, :)
       allocate(a(n, n))
       a = 0
       do i = 1, n
           a(i, 1) = 1
       end do
       do i = 2, n
           do j = 2, i
               a(i, j) = a(i - 1, j) + a(i - 1, j - 1)
           end do
       end do
   end function
   
   function pascal_upper(n) result(a)
       integer :: n, i, j
       integer, allocatable :: a(:, :)
       allocate(a(n, n))
       a = 0
       do i = 1, n
           a(1, i) = 1
       end do
       do i = 2, n
           do j = 2, i
               a(j, i) = a(j, i - 1) + a(j - 1, i - 1)
           end do
       end do
   end function
   function pascal_symmetric(n) result(a)
       integer :: n, i, j
       integer, allocatable :: a(:, :)
       allocate(a(n, n))
       a = 0
       do i = 1, n
           a(i, 1) = 1
           a(1, i) = 1
       end do
       do i = 2, n
           do j = 2, n
               a(i, j) = a(i - 1, j) + a(i, j - 1)
           end do
       end do
   end function
   subroutine print_matrix(a)
       integer :: a(:, :)
       integer :: n, i
       n = ubound(a, 1)
       do i = 1, n
           print *, a(i, :)
       end do
   end subroutine

end module

program ex_pascal

   use pascal
   implicit none
   integer :: n
   integer, allocatable :: a(:, :)
   print *, "Size?"
   read *, n
   print *, "Lower Pascal Matrix"
   a = pascal_lower(n)
   call print_matrix(a)
   print *, "Upper Pascal Matrix"
   a = pascal_upper(n)
   call print_matrix(a)
   print *, "Symmetric Pascal Matrix"
   a = pascal_symmetric(n)
   call print_matrix(a)

end program</lang>

<lang> Size? 5

Lower Pascal 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
Upper Pascal Matrix
          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
Symmetric Pascal Matrix
          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</lang>

FreeBASIC

<lang freebasic> sub print_matrix( M() as integer )

   'displays a matrix
   for row as integer = 0 to ubound(M, 1)
       for col as integer = 0 to ubound(M, 2)
           print using "####  ";M(row, col);
       next col
       print
   next row
   return

end sub

function fact( n as uinteger ) as uinteger

   'quick and dirty factorial
   if n<2 then return 1 else return n*fact(n-1)

end function

function nCp( n as uinteger, p as uinteger ) as uinteger

   'quick and dirty binomial
   if p>n then return 0 else return fact(n)/(fact(p)*fact(n-p))

end function

sub make_pascal( M() as integer, typ as const ubyte )

   'allocate the matrix first
   'typ 0 = jCi, 1=iCj, 2=(j+i)Ci
   for i as uinteger = 0 to ubound(M,1)
       for j as uinteger = 0 to ubound(M,2)
           select case typ
               case 0
                   M(i,j) = nCp(j, i)
               case 1
                   M(i,j) = nCp(i, j)
               case 2
                   M(i,j) = nCp(i + j, j)
               case else
                   M(i, j) = 0
           end select
       next j
   next i
   return

end sub

dim as integer M(0 to 4, 0 to 4) print "Upper triangular" make_pascal( M(), 0 ) print_matrix( M() ) print "Lower triangular" make_pascal( M(), 1 ) print_matrix( M() ) print "Symmetric" make_pascal( M(), 2 ) print_matrix( M() ) print "Technically the matrix needn't be square :)" dim as integer Q(0 to 4, 0 to 9) make_pascal( Q(), 2 ) print_matrix( Q() )</lang>

Output:

Upper triangular

  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  

Lower triangular

  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  

Symmetric

  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  

Technically the matrix needn't be square :)

  1     1     1     1     1     1     1     1     1     1  
  1     2     3     4     5     6     7     8     9    10  
  1     3     6    10    15    21    28    36    45    55  
  1     4    10    20    35    56    84   120   165   220  
  1     5    15    35    70   126   210   330   495   715

Go

Translation of: Kotlin

<lang go>package main

import (

   "fmt"
   "strings"

)

func binomial(n, k int) int {

   if n < k {
       return 0
   }
   if n == 0 || k == 0 {
       return 1
   }
   num := 1
   for i := k + 1; i <= n; i++ {
       num *= i
   }
   den := 1
   for i := 2; i <= n-k; i++ {
       den *= i
   }
   return num / den

}

func pascalUpperTriangular(n int) [][]int {

   m := make([][]int, n)
   for i := 0; i < n; i++ {
       m[i] = make([]int, n)
       for j := 0; j < n; j++ {
           m[i][j] = binomial(j, i)
       }
   }
   return m

}

func pascalLowerTriangular(n int) [][]int {

   m := make([][]int, n)
   for i := 0; i < n; i++ {
       m[i] = make([]int, n)
       for j := 0; j < n; j++ {
           m[i][j] = binomial(i, j)
       }
   }
   return m

}

func pascalSymmetric(n int) [][]int {

   m := make([][]int, n)
   for i := 0; i < n; i++ {
       m[i] = make([]int, n)
       for j := 0; j < n; j++ {
           m[i][j] = binomial(i+j, i)
       }
   }
   return m

}

func printMatrix(title string, m [][]int) {

   n := len(m)
   fmt.Println(title)
   fmt.Print("[")
   for i := 0; i < n; i++ {
       if i > 0 {
           fmt.Print(" ")
       }
       mi := strings.Replace(fmt.Sprint(m[i]), " ", ", ", -1)
       fmt.Print(mi)
       if i < n-1 {
           fmt.Println(",")
       } else {
           fmt.Println("]\n")
       }
   }

}

func main() {

   printMatrix("Pascal upper-triangular matrix", pascalUpperTriangular(5))
   printMatrix("Pascal lower-triangular matrix", pascalLowerTriangular(5))
   printMatrix("Pascal symmetric matrix", pascalSymmetric(5))

}</lang>

Output:
Pascal upper-triangular matrix
[[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]]

Pascal lower-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]]

Pascal symmetric matrix
[[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]]

Haskell

<lang haskell>import Data.List (transpose) import System.Environment (getArgs) import Text.Printf (printf)

-- Pascal's triangle. pascal :: Int pascal = iterate (\row -> 1 : zipWith (+) row (tail row) ++ [1]) [1]

-- The n by n Pascal lower triangular matrix. pascLow :: Int -> Int pascLow n = zipWith (\row i -> row ++ replicate (n-i) 0) (take n pascal) [1..]

-- The n by n Pascal upper triangular matrix. pascUp :: Int -> Int pascUp = transpose . pascLow

-- The n by n Pascal symmetric matrix. pascSym :: Int -> Int pascSym n = take n . map (take n) . transpose $ pascal

-- Format and print a matrix. printMat :: String -> Int -> IO () printMat title mat = do

 putStrLn $ title ++ "\n"
 mapM_ (putStrLn . concatMap (printf " %2d")) mat
 putStrLn "\n"

main :: IO () main = do

 ns <- fmap (map read) getArgs
 case ns of
   [n] -> do printMat "Lower triangular" $ pascLow n
             printMat "Upper triangular" $ pascUp  n
             printMat "Symmetric"        $ pascSym n
   _   -> error "Usage: pascmat <number>"</lang>
Output:
Lower triangular

  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


Upper triangular

  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


Symmetric

  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


Or, in terms of binomial coefficients and coordinate transformations:

<lang haskell>import Control.Monad (join) import Data.Bifunctor (bimap) import Data.Ix (range) import Data.List.Split (chunksOf) import Data.Tuple (swap)


PASCAL MATRIX ---------------------

pascalMatrix :: ((Int, Int) -> (Int, Int)) -> Int -> [Int] pascalMatrix f n =

 bc . f
   <$> range
     ((0, 0), join bimap pred (n, n))

-- Binomial coefficient bc :: (Int, Int) -> Int bc (n, k) =

 foldr
   (\x a -> quot (a * succ (n - x)) x)
   1
   [k, pred k .. 1]



TEST -------------------------

matrixSize = 5 :: Int

main :: IO () main =

 mapM_
   putStrLn
   ( unlines
       . ( \(s, xs) ->
             s :
             (show <$> chunksOf matrixSize xs)
         )
       <$> zip
         ["Lower", "Upper", "Symmetric"]
         ( pascalMatrix
             <$> [ id, -- Lower
                   swap, -- Upper
                   \(a, b) -> (a + b, b) -- Symmetric
                 ]
             <*> [matrixSize]
         )
   )</lang>
Output:
Lower
[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]

Upper
[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]

Symmetric
[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]

J

<lang J>  !/~ i. 5 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

  !~/~ i. 5

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

  (["0/ ! +/)~ i. 5

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</lang>

Explanation:

x!y is the number of ways of picking x balls (unordered) from a bag of y balls and x!/y for list x and list y gives a table where rows correspond to the elements of x and the columns correspond to the elements of y. Meanwhile !/~y is equivalent to y!/y (and i.y just counts the first y non-negative integers).

Also, x!~y is y!x (and the second example otherwise follows the same pattern as the first example.

For the final example we use an unadorned ! but prepare tables for its x and y values. Its right argument is a sum table, and its left argument is a left identity table. They look like this:

<lang J> (+/)~ i. 5 0 1 2 3 4 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 4 5 6 7 8

  (["0/)~ i. 5

0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4</lang>

The parenthesis in these last two examples are redundant - they could have been omitted without changing the result, but were left in place for emphasis.

Java

Translation of Python via D

Works with: Java version 8

<lang java>import static java.lang.System.out; import java.util.List; import java.util.function.Function; import java.util.stream.*; import static java.util.stream.Collectors.toList; import static java.util.stream.IntStream.range;

public class PascalMatrix {

   static int binomialCoef(int n, int k) {
       int result = 1;
       for (int i = 1; i <= k; i++)
           result = result * (n - i + 1) / i;
       return result;
   }
   static List<IntStream> pascal(int n, Function<Integer, IntStream> f) {
       return range(0, n).mapToObj(i -> f.apply(i)).collect(toList());
   }
   static List<IntStream> pascalUpp(int n) {
       return pascal(n, i -> range(0, n).map(j -> binomialCoef(j, i)));
   }
   static List<IntStream> pascalLow(int n) {
       return pascal(n, i -> range(0, n).map(j -> binomialCoef(i, j)));
   }
   static List<IntStream> pascalSym(int n) {
       return pascal(n, i -> range(0, n).map(j -> binomialCoef(i + j, i)));
   }
   static void print(String label, List<IntStream> result) {
       out.println("\n" + label);
       for (IntStream row : result) {
           row.forEach(i -> out.printf("%2d ", i));
           System.out.println();
       }
   }
   public static void main(String[] a) {
       print("Upper: ", pascalUpp(5));
       print("Lower: ", pascalLow(5));
       print("Symmetric:", pascalSym(5));
   }

}</lang>

Upper: 
 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 

Lower: 
 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 

Symmetric:
 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 

JavaScript

In terms of a binomial coefficient, and a function on a coordinate pair.

Translation of: Haskell

ES6

<lang JavaScript>(() => {

   'use strict';
   // -------------------PASCAL MATRIX--------------------
   // pascalMatrix :: ((Int, Int) -> (Int, Int)) ->
   // Int -> [Int]
   const pascalMatrix = f =>
       n => map(compose(binomialCoefficient, f))(
           range([0, 0], [n - 1, n - 1])
       );
   // binomialCoefficient :: (Int, Int) -> Int
   const binomialCoefficient = nk => {
       const [n, k] = Array.from(nk);
       return enumFromThenTo(k)(
           pred(k)
       )(1).reduceRight((a, x) => quot(
           a * succ(n - x)
       )(x), 1);
   };
   // ------------------------TEST------------------------
   // main :: IO ()
   const main = () => {
       const matrixSize = 5;
       console.log(intercalate('\n\n')(
           zipWith(
               k => xs => k + ':\n' + showMatrix(matrixSize)(xs)
           )(['Lower', 'Upper', 'Symmetric'])(
               apList(
                   map(pascalMatrix)([
                       identity, //              Lower
                       swap, //                  Upper
                       ([a, b]) => [a + b, b] // Symmetric
                   ])
               )([matrixSize])
           )
       ));
   };
   // ----------------------DISPLAY-----------------------
   // showMatrix :: Int -> [Int] -> String
   const showMatrix = n =>
       xs => {
           const
               ks = map(str)(xs),
               w = maximum(map(length)(ks));
           return unlines(
               map(unwords)(chunksOf(n)(
                   map(justifyRight(w)(' '))(ks)
               ))
           );
       };
   // -----------------GENERIC FUNCTIONS------------------
   // Tuple (,) :: a -> b -> (a, b)
   const Tuple = a =>
       b => ({
           type: 'Tuple',
           '0': a,
           '1': b,
           length: 2
       });
   // apList (<*>) :: [(a -> b)] -> [a] -> [b]
   const apList = fs =>
       // The sequential application of each of a list
       // of functions to each of a list of values.
       xs => fs.flatMap(
           f => xs.map(f)
       );
   // chunksOf :: Int -> [a] -> a
   const chunksOf = n =>
       xs => enumFromThenTo(0)(n)(
           xs.length - 1
       ).reduce(
           (a, i) => a.concat([xs.slice(i, (n + i))]),
           []
       );
   // compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
   const compose = (...fs) =>
       x => fs.reduceRight((a, f) => f(a), x);
   // concat :: a -> [a]
   // concat :: [String] -> String
   const concat = xs =>
       0 < xs.length ? (
           xs.every(x => 'string' === typeof x) ? (
               
           ) : []
       ).concat(...xs) : xs;
   // cons :: a -> [a] -> [a]
   const cons = x =>
       xs => [x].concat(xs);
   // enumFromThenTo :: Int -> Int -> Int -> [Int]
   const enumFromThenTo = x1 =>
       x2 => y => {
           const d = x2 - x1;
           return Array.from({
               length: Math.floor(y - x2) / d + 2
           }, (_, i) => x1 + (d * i));
       };
   // enumFromTo :: Int -> Int -> [Int]
   const enumFromTo = m =>
       n => Array.from({
           length: 1 + n - m
       }, (_, i) => m + i);
   // fst :: (a, b) -> a
   const fst = tpl =>
       // First member of a pair.
       tpl[0];
   // identity :: a -> a
   const identity = x =>
       // The identity function. (`id`, in Haskell)
       x;
   // intercalate :: String -> [String] -> String
   const intercalate = s =>
       // The concatenation of xs
       // interspersed with copies of s.
       xs => xs.join(s);
   // justifyRight :: Int -> Char -> String -> String
   const justifyRight = n =>
       // The string s, preceded by enough padding (with
       // the character c) to reach the string length n.
       c => s => n > s.length ? (
           s.padStart(n, c)
       ) : s;
   // length :: [a] -> Int
   const length = xs =>
       // Returns Infinity over objects without finite
       // length. This enables zip and zipWith to choose
       // the shorter argument when one is non-finite,
       // like cycle, repeat etc
       (Array.isArray(xs) || 'string' === typeof xs) ? (
           xs.length
       ) : Infinity;


   // liftA2List :: (a -> b -> c) -> [a] -> [b] -> [c]
   const liftA2List = f => xs => ys =>
       // The binary operator f lifted to a function over two
       // lists. f applied to each pair of arguments in the
       // cartesian product of xs and ys.
       xs.flatMap(
           x => ys.map(f(x))
       );
   // map :: (a -> b) -> [a] -> [b]
   const map = f =>
       // The list obtained by applying f to each element of xs.
       // (The image of xs under f).
       xs => (Array.isArray(xs) ? (
           xs
       ) : xs.split()).map(f);
   // maximum :: Ord a => [a] -> a
   const maximum = xs =>
       // The largest value in a non-empty list.
       0 < xs.length ? (
           xs.slice(1).reduce(
               (a, x) => x > a ? (
                   x
               ) : a, xs[0]
           )
       ) : undefined;
   // pred :: Enum a => a -> a
   const pred = x =>
       x - 1;
   // quot :: Int -> Int -> Int
   const quot = n => m => Math.floor(n / m);
   // The list of values in the subrange defined by a bounding pair.
   // range([0, 2]) -> [0,1,2]
   // range([[0,0], [2,2]])
   //  -> [[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2]]
   // range([[0,0,0],[1,1,1]])
   //  -> [[0,0,0],[0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]
   // range :: Ix a => (a, a) -> [a]
   function range() {
       const
           args = Array.from(arguments),
           ab = 1 !== args.length ? (
               args
           ) : args[0],
           [as, bs] = [ab[0], ab[1]].map(
               x => Array.isArray(x) ? (
                   x
               ) : (undefined !== x.type) &&
               (x.type.startsWith('Tuple')) ? (
                   Array.from(x)
               ) : [x]
           ),
           an = as.length;
       return (an === bs.length) ? (
           1 < an ? (
               traverseList(x => x)(
                   as.map((_, i) => enumFromTo(as[i])(bs[i]))
               )
           ) : enumFromTo(as[0])(bs[0])
       ) : [];
   };
   // snd :: (a, b) -> b
   const snd = tpl => tpl[1];
   // str :: a -> String
   const str = x => x.toString();
   // succ :: Enum a => a -> a
   const succ = x =>
       1 + x;
   // swap :: (a, b) -> (b, a)
   const swap = ab =>
       // The pair ab with its order reversed.
       Tuple(ab[1])(
           ab[0]
       );
   // take :: Int -> [a] -> [a]
   // take :: Int -> String -> String
   const take = n =>
       // The first n elements of a list,
       // string of characters, or stream.
       xs => xs.slice(0, n);
   // traverseList :: (Applicative f) => (a -> f b) -> [a] -> f [b]
   const traverseList = f =>
       // Collected results of mapping each element
       // of a structure to an action, and evaluating
       // these actions from left to right.
       xs => 0 < xs.length ? (() => {
           const
               vLast = f(xs.slice(-1)[0]),
               t = vLast.type || 'List';
           return xs.slice(0, -1).reduceRight(
               (ys, x) => liftA2List(cons)(f(x))(ys),
               liftA2List(cons)(vLast)([
                   []
               ])
           );
       })() : [
           []
       ];
   // unlines :: [String] -> String
   const unlines = xs =>
       // A single string formed by the intercalation
       // of a list of strings with the newline character.
       xs.join('\n');
   // unwords :: [String] -> String
   const unwords = xs =>
       // A space-separated string derived
       // from a list of words.
       xs.join(' ');
   // zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
   const zipWith = f =>
       // A list constructed by zipping with a
       // custom function, rather than with the
       // default tuple constructor.
       xs => ys => {
           const
               lng = Math.min(length(xs), length(ys)),
               vs = take(lng)(ys);
           return take(lng)(xs)
               .map((x, i) => f(x)(vs[i]));
       };
   // MAIN ---
   return main();

})();</lang>

Output:
Lower:
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

Upper:
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

Symmetric:
 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

jq

Works with: jq version 1.4

<lang jq># Generic functions

  1. Note: 'transpose' is defined in recent versions of jq

def transpose:

 if (.[0] | length) == 0 then []
 else [map(.[0])] + (map(.[1:]) | transpose)
 end ;
  1. Create an m x n matrix with init as the initial value

def matrix(m; n; init):

 if m == 0 then []
 elif m == 1 then [range(0;n) | init]
 elif m > 0 then
   matrix(1;n;init) as $row
   | [range(0;m) | $row ]
 else error("matrix\(m);_;_) invalid")
 end ;
  1. A simple pretty-printer for a 2-d matrix

def pp:

 def pad(n): tostring | (n - length) * " " + .;
 def row: reduce .[] as $x (""; . + ($x|pad(4)));
 reduce .[] as $row (""; . + "\n\($row|row)");</lang>

<lang jq># n is input def pascal_upper:

   . as $n
   | matrix($n; $n; 0)
   | .[0] = [range(0; $n) | 1 ] 
   | reduce range(1; $n) as $i
       (.; reduce range($i; $n) as $j
             (.; .[$i][$j] = .[$i-1][$j-1] + .[$i][$j-1]) ) ;

def pascal_lower:

 pascal_upper | transpose ;
  1. n is input

def pascal_symmetric:

   . as $n
   | matrix($n; $n; 1)
   | reduce range(1; $n) as $i
       (.; reduce range(1; $n) as $j
             (.; .[$i][$j] = .[$i-1][$j] + .[$i][$j-1]) ) ;</lang>

Example: <lang jq>5 | ("\nUpper:", (pascal_upper | pp),

  "\nLower:", (pascal_lower | pp),
  "\nSymmetric:", (pascal_symmetric | pp)
  )</lang>
Output:

<lang sh>$ jq -r -n -f Pascal_matrix_generation.jq

Upper:

  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

Lower:

  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

Symmetric:

  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</lang>

Julia

Julia has a built-in binomial function to compute the binomial coefficients, and we can construct the Pascal matrices with this function using list comprehensions: <lang Julia>julia> [binomial(j,i) for i in 0:4, j in 0:4] 5×5 Array{Int64,2}:

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

julia> [binomial(i,j) for i in 0:4, j in 0:4] 5×5 Array{Int64,2}:

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

julia> [binomial(j+i,i) for i in 0:4, j in 0:4] 5×5 Array{Int64,2}:

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

</lang>

Kotlin

<lang scala>// version 1.1.3

fun binomial(n: Int, k: Int): Int {

   if (n < k) return 0 
   if (n == 0 || k == 0) return 1
   val num = (k + 1..n).fold(1) { acc, i -> acc * i }
   val den = (2..n - k).fold(1) { acc, i -> acc * i }
   return num / den

}

fun pascalUpperTriangular(n: Int) = List(n) { i -> IntArray(n) { j -> binomial(j, i) } }

fun pascalLowerTriangular(n: Int) = List(n) { i -> IntArray(n) { j -> binomial(i, j) } }

fun pascalSymmetric(n: Int) = List(n) { i -> IntArray(n) { j -> binomial(i + j, i) } }

fun printMatrix(title: String, m: List<IntArray>) {

   val n = m.size
   println(title)
   print("[")
   for (i in 0 until n) {
       if (i > 0) print(" ")
       print(m[i].contentToString())
       if (i < n - 1) println(",") else println("]\n")
   }

}

fun main(args: Array<String>) {

   printMatrix("Pascal upper-triangular matrix", pascalUpperTriangular(5))
   printMatrix("Pascal lower-triangular matrix", pascalLowerTriangular(5))
   printMatrix("Pascal symmetric matrix", pascalSymmetric(5))

}</lang>

Output:
Pascal upper-triangular matrix
[[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]]

Pascal lower-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]]

Pascal symmetric matrix
[[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]]

Lua

<lang Lua>function factorial (n)

   local f = 1
   for i = 2, n do
       f = f * i
   end
   return f

end

function binomial (n, k)

   if k > n then return 0 end
   return factorial(n) / (factorial(k) * factorial(n - k))

end

function pascalMatrix (form, size)

   local matrix = {}
   for row = 1, size do
       matrix[row] = {}
       for col = 1, size do
           if form == "upper" then
               matrix[row][col] = binomial(col - 1, row - 1)
           end
           if form == "lower" then
               matrix[row][col] = binomial(row - 1, col - 1)
           end
           if form == "symmetric" then
               matrix[row][col] = binomial(row + col - 2, col - 1)
           end
       end
   end
   matrix.form = form:sub(1, 1):upper() .. form:sub(2, -1)
   return matrix

end

function show (mat)

   print(mat.form .. ":")
   for i = 1, #mat do
       for j = 1, #mat[i] do
           io.write(mat[i][j] .. "\t")
       end
       print()
   end
   print()

end

for _, form in pairs({"upper", "lower", "symmetric"}) do

   show(pascalMatrix(form, 5))

end</lang>

Output:
Upper:
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

Lower:
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

Symmetric:
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

Maple

<lang Maple> PascalUT := proc(n::integer)

local M := Matrix(n,n):
local i:
local j:
M[1,1..n] := 1:
for j from 2 to n do
 for i from 2 to n do
  M[i,j] := M[i,j-1] + M[i-1,j-1]:
 end:
end:
return M:

end proc:

PascalUT(5);

PascalLT := proc(n::integer)

local M := Matrix(n,n):
local i:
local j:
M[1..n,1] := 1:
for i from 2 to n do
 for j from 2 to n do
  M[i,j] := M[i-1,j] + M[i-1,j-1]:
 end:
end:
return M:

end proc:

PascalLT(5);

Pascal := proc(n::integer)

local M := Matrix(n,n):
local i:
local j:
M[1..n,1] := 1:
M[1,2..n] := 1:
for i from 2 to n do
 for j from 2 to n do
  M[i,j] := M[i,j-1] + M[i-1,j]:
 end:
end:
return M:

end proc:

Pascal(5);

</lang>

Output:

                           [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]
                           [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]
                         [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]

Mathematica /Wolfram Language

One solution is to generate a symmetric Pascal matrix then use the built in method to compute the upper Pascal matrix. This would be done as follows: <lang Mathematica>symPascal[size_] := NestList[Accumulate, Table[1, {k, size}], size - 1]

upperPascal[size_] := CholeskyDecomposition[symPascal@size]

lowerPascal[size_] := Transpose@CholeskyDecomposition[symPascal@size]

Column[MapThread[

 Labeled[Grid[#1@5], #2, Top] &, {{upperPascal, lowerPascal, 
   symPascal}, {"Upper", "Lower", "Symmetric"}}]]</lang>
Output:
Upper
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

Lower
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

Symmetric
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

It is also possible to directly compute a lower Pascal matrix as follows: <lang Mathematica>lowerPascal[size_] :=

MatrixExp[
 SparseArray[{Band[{2, 1}] -> Range[size - 1]}, {size, size}]]]</lang>

But since the builtin function MatrixExp works by first computing eigenvalues this is likely to be slower for large Pascal matrices

Nim

Using the function “binom” from module “math” of standard library. <lang Nim>import math, sequtils, strutils

type SquareMatrix = seq[seq[Natural]]

func newSquareMatrix(n: Positive): SquareMatrix =

 ## Create a square matrix.
 newSeqWith(n, newSeq[Natural](n))

func pascalUpperTriangular(n: Positive): SquareMatrix =

 ## Create an upper Pascal matrix.
 result = newSquareMatrix(n)
 for i in 0..<n:
   for j in i..<n:
     result[i][j] = binom(j, i)

func pascalLowerTriangular(n: Positive): SquareMatrix =

 ## Create a lower Pascal matrix.
 result = newSquareMatrix(n)
 for i in 0..<n:
   for j in i..<n:
     result[j][i] = binom(j, i)

func pascalSymmetric(n: Positive): SquareMatrix =

 ## Create a symmetric Pascal matrix.
 result = newSquareMatrix(n)
 for i in 0..<n:
   for j in 0..<n:
     result[i][j] = binom(i + j, i)

proc print(m: SquareMatrix) =

 ## Print a square matrix.
 let matMax = max(m.mapIt(max(it)))
 let length = ($matMax).len
 for i in 0..m.high:
   echo "| ", m[i].mapIt(($it).align(length)).join(" "), " |"

echo "Upper:" print pascalUpperTriangular(5) echo "\nLower:" print pascalLowerTriangular(5) echo "\nSymmetric:" print pascalSymmetric(5)</lang>

Output:
Upper:
| 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 |

Lower:
| 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 |

Symmetric:
|  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 |

PARI/GP

<lang parigp> Pl(n)={matpascal(n-1)} printf("%d",Pl(5)) </lang>

Output:
[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]

<lang parigp> Pu(n)={Pl(n)~} printf("%d",Pu(5)) </lang>

Output:
[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]

<lang parigp> Ps(n)={matrix(n,n,n,g,binomial(n+g-2,n-1))} printf("%d",Ps(5)) </lang>

Output:
[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]

Pascal

<lang pascal>program Pascal_matrix(Output);

const N = 5;

type NxN_Matrix = array[0..N,0..N] of integer;

var PM,PX : NxN_Matrix;

function Pascal_sym(x : integer; p : NxN_Matrix) : NxN_Matrix; var I,J : integer;

 begin
   for I := 1 to x do
   begin
     for J := 1 to x do p[I,J] := p[I-1,J]+p[I,J-1]
   end;
   Pascal_sym := p;
 end;

function Pascal_upp(x : integer; p : NxN_Matrix) : NxN_Matrix; var I,J : integer;

 begin
   for I := 1 to x do
   begin
     for J := 1 to x do p[I,J] := p[I-1,J-1]+p[I,J-1]
   end;
   Pascal_upp := p
 end;

function Pascal_low(x : integer; p : NxN_Matrix) : NxN_Matrix; var p1,p2 : NxN_Matrix;

 I,J : integer;
 begin
   p1 := Pascal_upp(x,p);
   p2 := p1;
   for I := 1 to x do
   begin
     for J := 1 to x do p1[J,I] := p2[I,J]
   end;
   Pascal_low := p1
 end;

procedure PrintMatrix(titel : ansistring; x : integer; p : NxN_Matrix); var I,J : integer;

 begin
   writeln(titel);
   for I := 1 to x do
   begin
     for J := 1 to x do write(p[I,J]:5);
     writeln();
   end;
 end;

begin

 PX[0,0] := 0;
 PM[0,0] := 1;
 PM := Pascal_upp(N, PM);
 PrintMatrix('Upper:', N, PM);
 writeln();
 PM := PX;
 PM[0,0] := 1;
 PM := Pascal_low(N, PM);
 PrintMatrix('Lower:', N, PM);
 writeln();
 PM := PX;
 PM[1,0] := 1;
 PM := Pascal_sym(N, PM);
 PrintMatrix('Symmetric', N, PM);
 writeln();
 readln;

end.</lang>

Output:
Upper:
    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

Lower:
    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

Symmetric
    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

Perl

<lang Perl>#!/usr/bin/perl use warnings; use strict; use feature qw{ say };


sub upper {

   my ($i, $j) = @_;
   my @m;
   for my $x (0 .. $i - 1) {
       for my $y (0 .. $j - 1) {
           $m[$x][$y] = $x > $y          ? 0
                      : ! $x || $x == $y ? 1
                                         : $m[$x-1][$y-1] + $m[$x][$y-1];
       }
   }
   return \@m

}


sub lower {

   my ($i, $j) = @_;
   my @m;
   for my $x (0 .. $i - 1) {
       for my $y (0 .. $j - 1) {
           $m[$x][$y] = $x < $y          ? 0
                      : ! $x || $x == $y ? 1
                                         : $m[$x-1][$y-1] + $m[$x-1][$y];
       }
   }
   return \@m

}


sub symmetric {

   my ($i, $j) = @_;
   my @m;
   for my $x (0 .. $i - 1) {
       for my $y (0 .. $j - 1) {
           $m[$x][$y] = ! $x || ! $y ? 1
                                     : $m[$x-1][$y] + $m[$x][$y-1];
       }
   }
   return \@m

}


sub pretty {

   my $m = shift;
   for my $row (@$m) {
       say join ', ', @$row;
   }

}


pretty(upper(5, 5)); say '-' x 14; pretty(lower(5, 5)); say '-' x 14; pretty(symmetric(5, 5));</lang>

Output:
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
--------------
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
--------------
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

Phix

Translation of: Fortran
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))
Output:
=== Pascal upper matrix ===
{{ 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}}
=== Pascal lower 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}}
=== Pascal symmetrical matrix ===
{{ 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}}

PicoLisp

<lang PicoLisp>(setq

  Low '(A B)
  Upp '(B A)
  Sym '((+ A B) A) )

(de binomial (N K)

  (let f
     '((N)
        (if (=0 N) 1 (apply * (range 1 N))) )
     (if (> K N)
        0
        (/
           (f N)
           (* (f (- N K)) (f K)) ) ) ) )

(de pascal (N Z)

  (for Lst
     (mapcar
        '((A)
           (mapcar
              '((B) (apply binomial (mapcar eval Z)))
              (range 0 N) ) )
        (range 0 N) )
     (for L Lst
        (prin (align 2 L) " ") )
     (prinl) )
  (prinl) )

(pascal 4 Low) (pascal 4 Upp) (pascal 4 Sym)</lang>

Output:
 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

 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

 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

PL/I

<lang pli> PASCAL_MATRIX: PROCEDURE OPTIONS (MAIN); /* derived from Fortran version 18 Decenber 2021 */

   pascal_lower: procedure(a);
       declare a(*,*) fixed binary;
       declare (n, i, j) fixed binary;
       n = hbound(a,1);
       a = 0;
       a(*, 1) = 1;
       do i = 2 to n;
           do j = 2 to i;
               a(i, j) = a(i - 1, j) + a(i - 1, j - 1);
           end;
       end;
   end pascal_lower;

   pascal_upper: procedure(a);
       declare a(*,*) fixed binary;
       declare (n, i, j) fixed binary;
       n = hbound(a,1);
       a = 0;
       a(1, *) = 1;
       do i = 2 to n;
           do j = 2 to i;
               a(j, i) = a(j, i - 1) + a(j - 1, i - 1);
           end;
       end;
   end pascal_upper;

   pascal_symmetric: procedure(a);
       declare a(*,*) fixed binary;
       declare (n, i, j) fixed binary;
       n = hbound(a,1);
       a = 0;
       a(*, 1) = 1;
       a(1, *) = 1;
       do i = 2 to n;
           do j = 2 to n;
               a(i, j) = a(i - 1, j) + a(i, j - 1);
           end;
       end;
   end pascal_symmetric;

   declare n fixed binary;
   put ('Size of matrix?');
   get (n);
   begin;
      declare a(n, n) fixed binary;
      put skip list ('Lower Pascal Matrix');
      call pascal_lower(a);
      put edit (a) (skip, (n) f(3) );
      put skip list ('Upper Pascal Matrix');
      call pascal_upper(a);
      put edit (a) (skip, (n) f(3) );
      put skip list ('Symmetric Pascal Matrix');
      call pascal_symmetric(a);
      put edit (a) (skip, (n) f(3) );
   end;

end PASCAL_MATRIX; </lang>

Output:
Size of matrix? 

Lower Pascal 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
Upper Pascal Matrix 
  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
Symmetric Pascal Matrix 
  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
Translation of: Rexx

<lang pli>*process source attributes xref or(!);

pat: Proc Options(main);
Dcl (HBOUND,MAX,RIGHT) Builtin;
Dcl SYSPRINT Print;
Dcl N Bin Fixed(31) Init(5);
Dcl pd Char(500) Var;
Dcl fact(0:10) Bin Fixed(31);
Dcl pt(0:500) Bin Fixed(31);
Call mk_fact(fact);
Call Pascal(n,'U',pt); Call show('Pascal upper triangular matrix');
Call Pascal(n,'L',pt); Call show('Pascal lower triangular matrix');
Call Pascal(n,'S',pt); Call show('Pascal symmetric matrix'       );
Pascal: proc(n,which,dd);
Dcl n Bin Fixed(31);
Dcl which Char(1);
Dcl (i,j,k) Bin Fixed(31);
Dcl dd(0:500) Bin Fixed(31);
k=0;
dd(0)=0;
do i=0 To n-1;
  Do j=0 To n-1;
    k+=1;
    Select(which);
      When('U') dd(k)=comb((j),  (i));
      When('L') dd(k)=comb((i),  (j));
      When('S') dd(k)=comb((i+j),(i));
      Otherwise;
      End;
    dd(0)=max(dd(0),dd(k));
    End;
  End;
End;
mk_fact: Proc(f);
Dcl f(0:*) Bin Fixed(31);
Dcl i Bin Fixed(31);
f(0)=1;
Do i=1 To hbound(f);
 f(i)=f(i-1)*i;
 End;
End;
comb: proc(x,y) Returns(pic'z9');
Dcl (x,y) Bin Fixed(31);
Dcl (j,z) Bin Fixed(31);
Dcl res Pic'Z9';
Select;
  When(x=y) res=1;
  When(y>x) res=0;
  Otherwise Do;
    If x-y<y then
      y=x-y;
    z=1;
    do j=x-y+1 to x;
      z=z*j;
      End;
    res=z/fact(y);
    End;
  End;
Return(res);
End;
show: Proc(head);
Dcl head Char(*);
Dcl (n,r,c,pl) Bin Fixed(31) Init(0);
Dcl row Char(50) Var;
Dcl p Pic'z9';
If pt(0)<10 Then pl=1;
            Else pl=2;
Dcl sep(5) Char(1) Init((4)(1)',',']');
Put Edit(' ',head)(Skip,a);
do r=1 To 5;
  if r=1 then row='[[';
         else row=' [';
  do c=1 To 5;
    n+=1;
    p=pt(n);
    row=row!!right(p,pl)!!sep(c);
    End;
  Put Edit(row)(Skip,a);
  End;
Put Edit(']')(A);
End;
End;</lang>
Output:
Pascal upper triangular matrix
[[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]]

Pascal lower 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]]

Pascal symmetric matrix
[[ 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]]

PureBasic

<lang PureBasic>EnableExplicit Define.i x=5, I, J

Macro Print_Pascal_matrix(typ)

 PrintN(typ)
 For I=1 To x
   For J=1 To x : Print(RSet(Str(p(I,J)),3," ")+Space(3)) : Next  
   PrintN("")
 Next
 Print(~"\n\n")  

EndMacro

Procedure Pascal_sym(n.i,Array p.i(2))

 Define.i I,J  
 p(1,0)=1
 For I=1 To n
   For J=1 To n : p(I,J)=p(I-1,J)+p(I,J-1) : Next
 Next

EndProcedure

Procedure Pascal_upp(n.i,Array p.i(2))

 Define.i I,J  
 p(0,0)=1
 For I=1 To n
   For J=1 To n : p(I,J)=p(I-1,J-1)+p(I,J-1) : Next
 Next  

EndProcedure

Procedure Pascal_low(n.i,Array p.i(2))

 Define.i I,J
 Pascal_upp(n,p())
 Dim p2.i(n,n)
 CopyArray(p(),p2())  
 For I=1 To n
   For J=1 To n : Swap p(J,I),p2(I,J) : Next
 Next  

EndProcedure

OpenConsole()

Dim p.i(x,x) Pascal_upp(x,p()) Print_Pascal_matrix("Upper:")

Dim p.i(x,x) Pascal_low(x,p()) Print_Pascal_matrix("Lower:")

Dim p.i(x,x) Pascal_sym(x,p()) Print_Pascal_matrix("Symmetric:")

Input() End</lang>

Output:
Upper:
  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


Lower:
  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


Symmetric:
  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

Python

Python: Procedural

Summing adjacent values: <lang python>from pprint import pprint as pp

def pascal_upp(n):

   s = [[0] * n for _ in range(n)]
   s[0] = [1] * n
   for i in range(1, n):
       for j in range(i, n):
           s[i][j] = s[i-1][j-1] + s[i][j-1]
   return s

def pascal_low(n):

   # transpose of pascal_upp(n)
   return [list(x) for x in zip(*pascal_upp(n))]

def pascal_sym(n):

   s = [[1] * n for _ in range(n)]
   for i in range(1, n):
       for j in range(1, n):
           s[i][j] = s[i-1][j] + s[i][j-1]
   return s
   

if __name__ == "__main__":

   n = 5
   print("\nUpper:")
   pp(pascal_upp(n))
   print("\nLower:")
   pp(pascal_low(n))
   print("\nSymmetric:")
   pp(pascal_sym(n))</lang>
Output:
Upper:
[[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]]

Lower:
[[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]]

Symmetric:
[[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]]


Using a binomial coefficient generator function:

<lang python>def binomialCoeff(n, k):

   result = 1
   for i in range(1, k+1):
       result = result * (n-i+1) // i
   return result

def pascal_upp(n):

   return [[binomialCoeff(j, i) for j in range(n)] for i in range(n)]

def pascal_low(n):

   return [[binomialCoeff(i, j) for j in range(n)] for i in range(n)]

def pascal_sym(n):

   return [[binomialCoeff(i+j, i) for j in range(n)] for i in range(n)]</lang>
Output:

(As above)

Python: Functional

Defining binomial coefficients in terms of reduce: <lang python>Pascal matrix generation

from functools import reduce from itertools import chain from operator import add


  1. pascalMatrix :: Int -> ((Int, Int) -> (Int, Int)) -> Int

def pascalMatrix(n):

   Pascal S-, L-, or U- matrix of order n.
   
   return lambda f: chunksOf(n)(list(map(
       compose(binomialCoefficent, f),
       tupleRange((0, 0), (n, n))
   )))


  1. binomialCoefficent :: (Int, Int) -> Int

def binomialCoefficent(nk):

   The binomial coefficient of the tuple (n, k).
   
   n, k = nk
   def go(a, x):
       return a * (n - x + 1) // x
   return reduce(go, enumFromTo(1)(k), 1)


  1. --------------------------TEST---------------------------
  2. main :: IO ()

def main():

   Pascal S-, L-, and U- matrices of order 5.
   
   order = 5
   for k, f in [
           ('Symmetric', lambda ab: (add(*ab), ab[1])),
           ('Lower', identity),
           ('Upper', swap)
   ]:
       print(k + ':')
       print(showMatrix(
           pascalMatrix(order)(f)
       ))
       print()


  1. --------------------REUSABLE GENERICS--------------------
  1. chunksOf :: Int -> [a] -> a

def chunksOf(n):

   A series of lists of length n, subdividing the
      contents of xs. Where the length of xs is not evenly
      divible, the final list will be shorter than n.
   
   return lambda xs: reduce(
       lambda a, i: a + [xs[i:n + i]],
       range(0, len(xs), n), []
   ) if 0 < n else []


  1. compose :: ((a -> a), ...) -> (a -> a)

def compose(*fs):

   Composition, from right to left,
      of a series of functions.
   
   return lambda x: reduce(
       lambda a, f: f(a),
       fs[::-1], x
   )


  1. enumFromTo :: Int -> Int -> [Int]

def enumFromTo(m):

   Enumeration of integer values [m..n]
   return lambda n: range(m, 1 + n)


  1. identity :: a -> a

def identity(x):

   The identity function.
   return x


  1. showMatrix :: Int -> String

def showMatrix(xs):

   String representation of xs
      as a matrix.
   
   def go():
       rows = [[str(x) for x in row] for row in xs]
       w = max(map(len, chain.from_iterable(rows)))
       return unlines(
           unwords(k.rjust(w, ' ') for k in row)
           for row in rows
       )
   return go() if xs else 


  1. swap :: (a, b) -> (b, a)

def swap(tpl):

   The swapped components of a pair.
   return tpl[1], tpl[0]


  1. tupleRange :: (Int, Int) -> (Int, Int) -> [(Int, Int)]

def tupleRange(lowerTuple, upperTuple):

   Range of (Int, Int) tuples from
      lowerTuple to upperTuple.
   
   l1, l2 = lowerTuple
   u1, u2 = upperTuple
   return [
       (i1, i2) for i1 in range(l1, u1)
       for i2 in range(l2, u2)
   ]


  1. unlines :: [String] -> String

def unlines(xs):

   A single string formed by the intercalation
      of a list of strings with the newline character.
   
   return '\n'.join(xs)


  1. unwords :: [String] -> String

def unwords(xs):

   A space-separated string derived
      from a list of words.
   
   return ' '.join(xs)


  1. MAIN ---

if __name__ == '__main__':

   main()</lang>
Output:
Symmetric:
 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

Lower:
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

Upper:
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

R

<lang r>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

}

  1. 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

}

  1. While it's possible to modify lower.pascal to get the upper matrix,
  2. 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

}</lang>

The results follow

<lang r>> lower.pascal(5)

    [,1] [,2] [,3] [,4] [,5]

[1,] 1 0 0 0 0 [2,] 1 1 0 0 0 [3,] 1 2 1 0 0 [4,] 1 3 3 1 0 [5,] 1 4 6 4 1 > lower.pascal.alt(5)

    [,1] [,2] [,3] [,4] [,5]

[1,] 1 0 0 0 0 [2,] 1 1 0 0 0 [3,] 1 2 1 0 0 [4,] 1 3 3 1 0 [5,] 1 4 6 4 1 > upper.pascal(5)

    [,1] [,2] [,3] [,4] [,5]

[1,] 1 1 1 1 1 [2,] 0 1 2 3 4 [3,] 0 0 1 3 6 [4,] 0 0 0 1 4 [5,] 0 0 0 0 1 > symm.pascal(5)

    [,1] [,2] [,3] [,4] [,5]

[1,] 1 1 1 1 1 [2,] 1 2 3 4 5 [3,] 1 3 6 10 15 [4,] 1 4 10 20 35 [5,] 1 5 15 35 70</lang>

Racket

<lang racket>#lang racket (require math/number-theory)

(define (pascal-upper-matrix n)

 (for/list ((i n)) (for/list ((j n)) (j . binomial . i))))

(define (pascal-lower-matrix n)

 (for/list ((i n)) (for/list ((j n)) (i . binomial . j))))

(define (pascal-symmetric-matrix n)

 (for/list ((i n)) (for/list ((j n)) ((+ i j) . binomial . j))))

(define (matrix->string m)

 (define col-width
   (for*/fold ((rv 1)) ((r m) (c r))
     (if (zero? c) rv (max rv (+ 1 (order-of-magnitude c))))))
 (string-append
  (string-join
  (for/list ((r m))
    (string-join (map (λ (c) (~a #:width col-width #:align 'right c)) r) " ")) "\n")
  "\n"))

(printf "Upper:~%~a~%" (matrix->string (pascal-upper-matrix 5))) (printf "Lower:~%~a~%" (matrix->string (pascal-lower-matrix 5))) (printf "Symmetric:~%~a~%" (matrix->string (pascal-symmetric-matrix 5)))</lang>

Output:
Upper:
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

Lower:
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

Symmetric:
 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

Raku

(formerly Perl 6)

Works with: rakudo version 2016-12

Here is a rather more general solution than required. The grow-matrix function will grow any N by N matrix into an N+1 x N+1 matrix, using any function of the three leftward/upward neighbors, here labelled "West", "North", and "Northwest". We then define three iterator functions that can grow Pascal matrices, and use those iterators to define three constants, each of which is an infinite sequence of ever-larger Pascal matrices. Normal subscripting then pulls out the ones of the specified size. <lang perl6># Extend a matrix in 2 dimensions based on 3 neighbors. sub grow-matrix(@matrix, &func) {

   my $n = @matrix.shape eq '*' ?? 1 !! @matrix.shape[0];
   my @m[$n+1;$n+1];
   for ^$n X ^$n -> ($i, $j) {
      @m[$i;$j] = @matrix[$i;$j];
   }
  1. West North NorthWest
   @m[$n; 0] = func( 0,           @m[$n-1;0],  0            );
   @m[ 0;$n] = func( @m[0;$n-1],  0,           0            );
   @m[$_;$n] = func( @m[$_;$n-1], @m[$_-1;$n], @m[$_-1;$n-1]) for 1 ..^ $n;
   @m[$n;$_] = func( @m[$n;$_-1], @m[$n-1;$_], @m[$n-1;$_-1]) for 1 ..  $n;
   @m;

}

  1. I am but mad north-northwest...

sub madd-n-nw(@m) { grow-matrix @m, -> $w, $n, $nw { $n + $nw } } sub madd-w-nw(@m) { grow-matrix @m, -> $w, $n, $nw { $w + $nw } } sub madd-w-n (@m) { grow-matrix @m, -> $w, $n, $nw { $w + $n } }

  1. Define 3 infinite sequences of Pascal matrices.

constant upper-tri = [1], &madd-w-nw ... *; constant lower-tri = [1], &madd-n-nw ... *; constant symmetric = [1], &madd-w-n ... *;

show_m upper-tri[4]; show_m lower-tri[4]; show_m symmetric[4];

sub show_m (@m) { my \n = @m.shape[0]; for ^n X ^n -> (\i, \j) {

   print @m[i;j].fmt("%{1+max(@m).chars}d"); 
   print "\n" if j+1 eq n;

} say ; }</lang>

Output:
 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

 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

  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

REXX

separate generation

Commentary:   1/3   of the REXX program deals with the displaying of the matrix. <lang rexx>/*REXX program generates and displays three forms of an NxN Pascal matrix. */ numeric digits 50 /*be able to calculate huge factorials.*/ parse arg N . /*obtain the optional matrix size (N).*/ if N== | N=="," then N= 5 /*Not specified? Then use the default.*/

                              call show  N,  upp(N),  'Pascal upper triangular matrix'
                              call show  N,  low(N),  'Pascal lower triangular matrix'
                              call show  N,  sym(N),  'Pascal symmetric matrix'

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ upp: procedure; parse arg N; $= /*gen Pascal upper triangular matrix. */

           do i=0  for N;  do j=0  for N; $=$ comb(j,   i);   end; end;   return $

/*──────────────────────────────────────────────────────────────────────────────────────*/ low: procedure; parse arg N; $= /*gen Pascal lower triangular matrix. */

           do i=0  for N;  do j=0  for N; $=$ comb(i,   j);   end; end;   return $

/*──────────────────────────────────────────────────────────────────────────────────────*/ sym: procedure; parse arg N; $= /*generate Pascal symmetric matrix. */

           do i=0  for N;  do j=0  for N; $=$ comb(i+j, i);   end; end;   return $

/*──────────────────────────────────────────────────────────────────────────────────────*/ !: procedure; parse arg x;  !=1; do j=2 to x;  != !*j; end; return ! /*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure; parse arg x,y; if x=y then return 1 /* {=} case.*/

                                      if y>x  then return 0                /* {>} case.*/
     if x-y<y  then y= x-y;  _= 1;    do j=x-y+1  to x;  _= _*j;  end;    return _ / !(y)

/*──────────────────────────────────────────────────────────────────────────────────────*/ show: procedure; parse arg s,@; w=0; #=0 /*get args. */

                      do x=1  for s**2;  w= max(w, 1 + length( word(@,x) ) );    end
     say;   say center( arg(3), 50, '─')                                   /*show title*/
                      do    r=1  for s;  if r==1  then $= '[['             /*row  1    */
                                                  else $= ' ['             /*rows 2   N*/
                         do c=1  for s;  #= #+1;   e= (c==s)               /*e ≡ "end".*/
                         $=$ || right( word(@, #), w) || left(',', \e) || left("]", e)
                         end   /*c*/                                       /* [↑]  row.*/
                      say $ || left(',', r\==s)left("]", r==s)             /*show row. */
                      end     /*r*/
     return</lang>
output   when using the default input:
──────────Pascal upper triangular matrix──────────
[[ 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]]

──────────Pascal lower 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]]

─────────────Pascal symmetric matrix──────────────
[[  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]]

consolidated generation

This REXX version uses a consolidated generation subroutine, even though this Rosetta Code implies to use   functions   (instead of a single function). <lang rexx>/*REXX program generates and displays three forms of an NxN Pascal matrix. */ numeric digits 50 /*be able to calculate huge factorials.*/ parse arg N . /*obtain the optional matrix size (N).*/ if N== | N=="," then N= 5 /*Not specified? Then use the default.*/

                          call show N, Pmat(N, 'upper'), 'Pascal upper triangular matrix'
                          call show N, Pmat(N, 'lower'), 'Pascal lower triangular matrix'
                          call show N, Pmat(N, 'sym')  , 'Pascal symmetric matrix'

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ Pmat: procedure; parse arg N; $= /*generate a format of a Pascal matrix.*/

     arg , ?                                    /*get uppercase version of the 2nd arg.*/
             do i=0  for N; do j=0  for N       /*pick a format to use  [↓]            */
                            if abbrev('UPPER'      , ?, 1)  then $= $ comb(j  , i)
                            if abbrev('LOWER'      , ?, 1)  then $= $ comb(i  , j)
                            if abbrev('SYMMETRICAL', ?, 1)  then $= $ comb(i+j, j)
                            end  /*j*/         /*       ↑                              */
             end   /*i*/                       /*       │                              */
     return $                                  /*       └──min. abbreviation is 1 char.*/

/*──────────────────────────────────────────────────────────────────────────────────────*/ !: procedure; parse arg x;  !=1; do j=2 to x;  != ! * j; end; return ! /*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure; parse arg x,y; if x=y then return 1 /* {=} case.*/

                                      if y>x  then return 0                /* {>} case.*/
     if x-y<y  then y=x-y; _= 1;      do j=x-y+1  to x;  _= _ * j;  end;  return _ / !(y)

/*──────────────────────────────────────────────────────────────────────────────────────*/ show: procedure; parse arg s,@; w=0; #=0 /*get args. */

                      do x=1  for s**2;  w=max(w,1+length(word(@,x)));  end
     say;   say center( arg(3), 50, '─')                                   /*show title*/
                      do   r=1  for s;   if r==1  then $= '[['             /*row  1    */
                                                  else $= ' ['             /*rows 2   N*/
                         do c=1  for s;  #= # + 1;     e= (c==s)           /*e ≡ "end".*/
                         $=$ || right( word(@, #), w) || left(', ',\e) || left("]", e)
                         end   /*c*/                                       /* [↑]  row.*/
                      say $ || left(',', r\==s)left(']', r==s)             /*show row. */
                      end     /*r*/
     return</lang>
output   is identical to the 1st REXX version.



Ring

<lang ring>

  1. Project : Pascal matrix generation

load "stdlib.ring" res = newlist(5,5)

see "=== Pascal upper matrix ===" + nl result = pascalupper(5) showarray(result)

see nl + "=== Pascal lower matrix ===" + nl result = pascallower(5) showarray(result)

see nl + "=== Pascal symmetrical matrix ===" + nl result = pascalsymmetric(5) showarray(result)

func pascalupper(n)

   for m=1 to n
         for p=1 to n
              res[m][p] = 0
         next
   next 
   for p=1 to n
        res[1][p] = 1
   next    
   for i=2 to n 
       for j=2 to i 
           res[j][i] = res[j][i-1]+res[j-1][i-1]
       end 
   end 
   return res

func pascallower(n)

       for m=1 to n
             for p=1 to n
                  res[m][p] = 0
             next
       next
      for p=1 to n  
            res[p][1] = 1
      next
      for i=2 to n 
           for j=2 to i 
                res[i][j] = res[i-1][j]+res[i-1][j-1]
           next
       next
       return res

func pascalsymmetric(n)

       for m=1 to n
             for p=1 to n
                  res[m][p] = 0
             next
       next
       for p=1 to n 
             res[p][1] = 1
             res[1][p] = 1
       next
       for i=2 to n 
            for j = 2 to n 
                 res[i][j] = res[i-1][j]+res[i][j-1]
            next
       next
       return res

func showarray(result)

       for n=1 to 5
             for m=1 to 5
                  see "" + result[n][m] + " "
             next
            see nl
       next

</lang> Output:

=== Pascal upper matrix ===
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 

=== Pascal lower 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 

=== Pascal symmetrical matrix ===
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 

Ruby

Summing adjacent values: <lang ruby>#Upper, lower, and symetric Pascal Matrix - Nigel Galloway: May 3rd., 21015 require 'pp'

ng = (g = 0..4).collect{[]} g.each{|i| g.each{|j| ng[i][j] = i==0 ? 1 : j<i ? 0 : ng[i-1][j-1]+ng[i][j-1]}} pp ng; puts g.each{|i| g.each{|j| ng[i][j] = j==0 ? 1 : i<j ? 0 : ng[i-1][j-1]+ng[i-1][j]}} pp ng; puts g.each{|i| g.each{|j| ng[i][j] = (i==0 or j==0) ? 1 : ng[i-1][j ]+ng[i][j-1]}} pp ng</lang>

Output:
[[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]]

[[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]]

[[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]]

Binomial coefficient:

<lang ruby>require 'pp'

def binomial_coeff(n,k) (1..k).inject(1){|res,i| res * (n-i+1) / i} end

def pascal_upper(n) (0...n).map{|i| (0...n).map{|j| binomial_coeff(j,i)}} end def pascal_lower(n) (0...n).map{|i| (0...n).map{|j| binomial_coeff(i,j)}} end def pascal_symmetric(n) (0...n).map{|i| (0...n).map{|j| binomial_coeff(i+j,j)}} end

puts "Pascal upper-triangular matrix:" pp pascal_upper(5)

puts "\nPascal lower-triangular matrix:" pp pascal_lower(5)

puts "\nPascal symmetric matrix:" pp pascal_symmetric(5)</lang>

Output:
Pascal upper-triangular matrix:
[[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]]

Pascal lower-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]]

Pascal symmetric matrix:
[[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]]

Scala

<lang scala>//Pascal Matrix Generator

object pascal{ def main( args:Array[String] ){

println("Enter the order of matrix") val n = scala.io.StdIn.readInt()

var F = new Factorial()

var mx = Array.ofDim[Int](n,n)

for( i <- 0 to (n-1); j <- 0 to (n-1) ){

if( i>=j ){ //iCj mx(i)(j) = F.fact(i) / ( ( F.fact(j) )*( F.fact(i-j) ) ) } }

println("iCj:") for( i <- 0 to (n-1) ){ //iCj print for( j <- 0 to (n-1) ){ print( mx(i)(j)+" " ) } println("") }

println("jCi:") for( i <- 0 to (n-1) ){ //jCi print for( j <- 0 to (n-1) ){ print( mx(j)(i)+" " ) } println("") }

//(i+j)C j for( i <- 0 to (n-1); j <- 0 to (n-1) ){

mx(i)(j) = F.fact(i+j) / ( ( F.fact(j) )*( F.fact(i) ) ) } //print (i+j)Cj println("(i+j)Cj:") for( i <- 0 to (n-1) ){ for( j <- 0 to (n-1) ){ print( mx(i)(j)+" " ) } println("") }

} }

class Factorial(){

def fact( a:Int ): Int = {

var b:Int = 1

for( i <- 2 to a ){ b = b*i } return b } } </lang>

Scheme

Using SRFI-25:

<lang scheme>(import (srfi 25))

(define-syntax dotimes

 (syntax-rules ()
   ((_ (i n) body ...)
    (do ((i 0 (+ i 1)))
        ((>= i n))
      body ...))))


(define (pascal-upper n)

 (let ((p (make-array (shape 0 n 0 n) 0)))
   (dotimes (i n)
     (array-set! p 0 i 1))
    (dotimes (i (- n 1))
      (dotimes (j (- n 1))
        (array-set! p (+ 1 i) (+ 1 j)
                    (+ (array-ref p i j)
                       (array-ref p (+ 1 i) j)))))
    p))

(define (pascal-lower n)

 (let ((p (make-array (shape 0 n 0 n) 0)))
   (dotimes (i n)
     (array-set! p i 0 1))
    (dotimes (i (- n 1))
      (dotimes (j (- n 1))
        (array-set! p (+ 1 i) (+ 1 j)
                    (+ (array-ref p i j)
                       (array-ref p i (+ 1 j))))))
    p))

(define (pascal-symmetric n)

 (let ((p (make-array (shape 0 n 0 n) 0)))
   (dotimes (i n)
     (array-set! p i 0 1)
     (array-set! p 0 i 1))
    (dotimes (i (- n 1))
      (dotimes (j (- n 1))
        (array-set! p (+ 1 i) (+ 1 j)
                    (+ (array-ref p (+ 1 i) j)
                       (array-ref p i (+ 1 j))))))
    p))


(define (print-array a)

 (let ((r (array-end a 0))
       (c (array-end a 1)))
   (dotimes (row (- r 1))
     (dotimes (col (- c 1))
       (display (array-ref a row col))
       (display #\space))
     (newline))))</lang>

<lang scheme>(print-array (pascal-upper 6))</lang>

Output:
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 

<lang scheme>(print-array (pascal-lower 6))</lang>

Output:
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 

<lang scheme>(print-array (pascal-symmetric 6))</lang>

Output:
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 

Sidef

Translation of: Raku

<lang ruby>func grow_matrix(matrix, callback) {

   var m = matrix
   var s = m.len
   m[s][0] = callback(0, m[s-1][0], 0)
   m[0][s] = callback(m[0][s-1], 0, 0)
return m } func transpose(matrix) { matrix[0].range.map{|i| matrix.map{_[i]} } } func madd_n_nw(m) { grow_matrix(m, ->(_, n, nw) { n + nw }) } func madd_w_nw(m) { grow_matrix(m, ->(w, _, nw) { w + nw }) } func madd_w_n(m) { grow_matrix(m, ->(w, n, _) { w + n }) } var functions = [madd_n_nw, madd_w_nw, madd_w_n].map { |f| func(n) { var r = { f(r) } * n transpose(r) } } functions.map { |f| f(4).map { .map{ '%2s' % _ }.join(' ') }.join("\n") }.join("\n\n").say</lang>
Output:
 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

 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

 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

Stata

Here are variants for the lower matrix.

<lang stata>mata function pascal1(n) { return(comb(J(1,n,0::n-1),J(n,1,0..n-1))) }

function pascal2(n) { a = I(n) a[.,1] = J(n,1,1) for (i=3; i<=n; i++) { a[i,2..i-1] = a[i-1,2..i-1]+a[i-1,1..i-2] } return(a) }

function pascal3(n) { a = J(n,n,0) for (i=1; i<n; i++) { a[i+1,i] = i } s = p = I(n) k = 1 for (i=0; i<n; i++) { p = p*a/k++ s = s+p } return(s) } end</lang>

One could trivially write functions for the upper matrix (same operations with transposed matrices). The symmetric matrix can be generated using loops. However, when the lower matrix is known the other two are immediately deduced:

<lang stata>: a = pascal3(5)

a
      1   2   3   4   5
   +---------------------+
 1 |  1   0   0   0   0  |
 2 |  1   1   0   0   0  |
 3 |  1   2   1   0   0  |
 4 |  1   3   3   1   0  |
 5 |  1   4   6   4   1  |
   +---------------------+
a'
      1   2   3   4   5
   +---------------------+
 1 |  1   1   1   1   1  |
 2 |  0   1   2   3   4  |
 3 |  0   0   1   3   6  |
 4 |  0   0   0   1   4  |
 5 |  0   0   0   0   1  |
   +---------------------+
a*a'

[symmetric]

       1    2    3    4    5
   +--------------------------+
 1 |   1                      |
 2 |   1    2                 |
 3 |   1    3    6            |
 4 |   1    4   10   20       |
 5 |   1    5   15   35   70  |
   +--------------------------+</lang>

The last is a symmetric matrix, but Stata only shows the lower triangular part of symmetric matrices.

Tcl

<lang Tcl> package require math

namespace eval pascal {

   proc upper {n} {
       for {set i 0} {$i < $n} {incr i} {
           for {set j 0} {$j < $n} {incr j} {
               puts -nonewline \t[::math::choose $j $i]
           }
           puts ""
       }
   }
   proc lower {n} {
       for {set i 0} {$i < $n} {incr i} {
           for {set j 0} {$j < $n} {incr j} {
               puts -nonewline \t[::math::choose $i $j]
           }
           puts ""
       }
   }
   proc symmetric {n} {
       for {set i 0} {$i < $n} {incr i} {
           for {set j 0} {$j < $n} {incr j} {
               puts -nonewline \t[::math::choose [expr {$i+$j}] $i]
           }
           puts ""
       }
   }

}

foreach type {upper lower symmetric} {

   puts "\n* $type"
   pascal::$type 5

} </lang>

Output:
* upper 
        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

* lower 
        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

* symmetric
        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

VBA

Translation of: Phix

<lang vb>Option Base 1 Private Function pascal_upper(n As Integer)

   Dim res As Variant: ReDim res(n, n)
   For j = 1 To n
       res(1, j) = 1
   Next j
   For i = 2 To n
       res(i, 1) = 0
       For j = 2 To i
           res(j, i) = res(j, i - 1) + res(j - 1, i - 1)
       Next j
       For j = i + 1 To n
           res(j, i) = 0
       Next j
   Next i
   pascal_upper = res

End Function

Private Function pascal_symmetric(n As Integer)

   Dim res As Variant: ReDim res(n, n)
   For i = 1 To n
       res(i, 1) = 1
       res(1, i) = 1
   Next i
   For i = 2 To n
       For j = 2 To n
           res(i, j) = res(i - 1, j) + res(i, j - 1)
       Next j
   Next i
   pascal_symmetric = res

End Function

Private Sub pp(m As Variant)

   For i = 1 To UBound(m)
       For j = 1 To UBound(m, 2)
           Debug.Print Format(m(i, j), "@@@");
       Next j
       Debug.Print
   Next i

End Sub

Public Sub main()

   Debug.Print "=== Pascal upper matrix ==="
   pp pascal_upper(5)
   Debug.Print "=== Pascal lower matrix ==="
   pp WorksheetFunction.Transpose(pascal_upper(5))
   Debug.Print "=== Pascal symmetrical matrix ==="
   pp pascal_symmetric(5)
End Sub</lang>
Output:
=== Pascal upper matrix ===
  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
=== Pascal lower 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
=== Pascal symmetrical matrix ===
  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

VBScript

<lang vb> Function pascal_upper(i,j) WScript.StdOut.Write "Pascal Upper" WScript.StdOut.WriteLine For l = i To j For m = i To j If l <= m Then WScript.StdOut.Write binomial(m,l) & vbTab Else WScript.StdOut.Write 0 & vbTab End If Next WScript.StdOut.WriteLine Next WScript.StdOut.WriteLine End Function

Function pascal_lower(i,j) WScript.StdOut.Write "Pascal Lower" WScript.StdOut.WriteLine For l = i To j For m = i To j If l >= m Then WScript.StdOut.Write binomial(l,m) & vbTab Else WScript.StdOut.Write 0 & vbTab End If Next WScript.StdOut.WriteLine Next WScript.StdOut.WriteLine End Function

Function pascal_symmetric(i,j) WScript.StdOut.Write "Pascal Symmetric" WScript.StdOut.WriteLine For l = i To j For m = i To j WScript.StdOut.Write binomial(l+m,m) & vbTab Next WScript.StdOut.WriteLine Next End Function

Function binomial(n,k) binomial = factorial(n)/(factorial(n-k)*factorial(k)) End Function

Function factorial(n) If n = 0 Then factorial = 1 Else For i = n To 1 Step -1 If i = n Then factorial = n Else factorial = factorial * i End If Next End If End Function

'Test driving Call pascal_upper(0,4) Call pascal_lower(0,4) Call pascal_symmetric(0,4) </lang>

Output:
Pascal Upper
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	

Pascal Lower
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	

Pascal Symmetric
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	

Wren

Library: Wren-fmt
Library: Wren-math
Library: Wren-matrix

<lang ecmascript>import "/fmt" for Fmt import "/math" for Int import "/matrix" for Matrix

var binomial = Fn.new { |n, k|

   if (n == k) return 1
   var prod = 1
   var i = n - k + 1
   while (i <= n) {
       prod = prod * i
       i = i + 1
   }
   return prod / Int.factorial(k)

}

var pascalUpperTriangular = Fn.new { |n|

   var m = List.filled(n, null)
   for (i in 0...n) {
       m[i] = List.filled(n, 0)
       for (j in 0...n) m[i][j] = binomial.call(j, i)
   }
   return Matrix.new(m)

}

var pascalSymmetric = Fn.new { |n|

   var m = List.filled(n, null)
   for (i in 0...n) {
       m[i] = List.filled(n, 0)
       for (j in 0...n) m[i][j] = binomial.call(i+j, i)
   }
   return Matrix.new(m)

}

var pascalLowerTriangular = Fn.new { |n| pascalSymmetric.call(n).cholesky() }

var n = 5 System.print("Pascal upper-triangular matrix:") Fmt.mprint(pascalUpperTriangular.call(n), 2, 0) System.print("\nPascal lower-triangular matrix:") Fmt.mprint(pascalLowerTriangular.call(n), 2, 0) System.print("\nPascal symmetric matrix:") Fmt.mprint(pascalSymmetric.call(n), 2, 0)</lang>

Output:
Pascal upper-triangular matrix:
| 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|

Pascal lower-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|

Pascal symmetric matrix:
| 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|

zkl

Translation of: Python

<lang zkl>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)

}</lang> <lang zkl>fcn prettyPrint(m){ // m is a list of lists

  fmt:=("%3d "*m.len() + "\n").fmt;
  m.pump(String,'wrap(col){ fmt(col.xplode()) });

} const N=5; println("Upper:\n", pascal_upp(N):prettyPrint(_)); println("Lower:\n", pascal_low(N):prettyPrint(_)); println("Symmetric:\n",pascal_sym(N):prettyPrint(_));</lang>

Output:
Upper:
  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 

Lower:
  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 

Symmetric:
  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 

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