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{{task}}
A pascal matrix is a two-dimensional square matrix holding numbers from [[Pascal's triangle]], also known as [[Evaluate binomial coefficients|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.
360 Assembly
* 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
{{out}}
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
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;
{{out}}
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
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
{{out}}
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
{{Trans|ALGOL_68}}
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.
{{out}}
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
AppleScript
By composition of generic functions:
-- 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
{{Out}}
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]
C
#include <stdio.h>
#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;
}
{{out}}
### 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++
{{works with|GCC|version 7.2.0 (Ubuntu 7.2.0-8ubuntu3.2) }}
#include <iostream>
#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));
}
{{out}}
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#
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;
}
}
{{out}}
|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|
Common 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)
#2A((1 0 0 0) (1 1 0 0) (1 2 1 0) (1 3 3 1))
? (pascal-upper 4)
#2A((1 1 1 1) (0 1 2 3) (0 0 1 3) (0 0 0 1))
? (pascal-symmetric 4)
#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
D
{{trans|Python}}
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));
}
{{out}}
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
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<i -> 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)
{{out}}
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]
Factor
{{works with|Factor|0.99}}
USING: arrays fry io kernel math math.combinatorics
math.matrices prettyprint sequences ;
IN: rosetta-code.pascal-matrix
: pascal ( n quot -- m )
'[
dup 2array matrix-coordinates [ first2 @ nCk ]
matrix-map
] call ; inline
: lower ( n -- m ) [ ] pascal ;
: upper ( n -- m ) lower flip ;
: symmetric ( n -- m ) [ dup [ + ] dip ] pascal ;
: pascal-matrix-demo ( -- )
5 [ lower "Lower:" ] [ upper "Upper:" ]
[ symmetric "Symmetric:" ] tri
[ print simple-table. nl ] 2tri@ ;
MAIN: pascal-matrix-demo
{{out}}
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
Fortran
The following program uses features of Fortran 2003.
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
## Go
{{trans|Kotlin}}
```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))
}
{{out}}
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
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>"
{{out}}
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:
import Data.Ix (range)
import Data.Tuple (swap)
import Data.List.Split (chunksOf)
-- (Transform on coordinate pair) -> Matrix size -> Matrix values
pascalMatrix :: ((Int, Int) -> (Int, Int)) -> Int -> [Int]
pascalMatrix f n = (bc . f) <$> range ((0, 0), (n - 1, n - 1))
-- Binomial coefficient
bc :: (Int, Int) -> Int
bc (n, k) = foldr (\x a -> quot (a * (n - x + 1)) x) 1 [k,k - 1 .. 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]))
{{Out}}
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
!/~ 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
Explanation:
x!y is [http://www.jsoftware.com/jwiki/Vocabulary/bang#dyadic 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:
(+/)~ 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
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 [[Pascal_matrix_generation#Python|Python]] via [[Pascal_matrix_generation#D|D]] {{works with|Java|8}}
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));
}
}
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. {{Trans|Haskell}}
ES6
(() => {
'use strict';
// PASCAL MATRIX ---------------------------------------------------------
// (Function on a coordinate pair) -> Matrix size -> Matrix rows
// pascalMatrix :: ((Int, Int) -> (Int, Int)) -> Int -> [[Int]]
const pascalMatrix = (f, n) =>
chunksOf(n, map(compose(bc, f), range([
[0, 0],
[n - 1, n - 1]
])));
// Binomial coefficient
// bc :: (Int, Int) -> Int
const bc = ([n, k]) => {
return enumFromTo(1, k)
.reduce((a, x) => {
return Math.floor((a * (n - x + 1)) / x)
}, 1)
};
// GENERIC FUNCTIONS -----------------------------------------------------
// A list of functions applied to a list of arguments
// <*> :: [(a -> b)] -> [a] -> [b]
const ap = (fs, xs) => //
[].concat.apply([], fs.map(f => //
[].concat.apply([], xs.map(x => [f(x)]))));
// chunksOf :: Int -> [a] -> [[a]]
const chunksOf = (n, xs) =>
xs.reduce((a, _, i, xs) =>
i % n ? a : a.concat([xs.slice(i, i + n)]), []);
// compose :: (b -> c) -> (a -> b) -> (a -> c)
const compose = (f, g) => x => f(g(x));
// concat :: [[a]] -> [a] | [String] -> String
const concat = xs => {
if (xs.length > 0) {
const unit = typeof xs[0] === 'string' ? '' : [];
return unit.concat.apply(unit, xs);
} else return [];
};
// cons :: a -> [a] -> [a]
const cons = (x, xs) => [x].concat(xs);
// curry :: ((a, b) -> c) -> a -> b -> c
const curry = f => a => b => f(a, b);
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);
// id :: a -> a
const id = x => x;
// log :: a -> IO ()
const log = (...args) =>
console.log(
args
.map(JSON.stringify)
.join(' -> ')
);
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);
// range :: Ix a => (a, a) -> [a]
const range = ([a, b]) => {
const [as, bs] = a instanceof Array ? [a, b] : [
[a],
[b]
],
an = as.length;
return (an === bs.length) ? (
an > 1 ? (
sequence(as.map((_, i) => enumFromTo(as[i], bs[i])))
) : enumFromTo(a, b)
) : [];
};
// Evaluate left to right, and collect the results
// sequence :: Monad m => [m a] -> m [a]
const sequence = xs => traverse(id, xs);
// show ::
// (a -> String) f, Num n =>
// a -> maybe f -> maybe n -> String
const show = JSON.stringify;
// swap :: (a, b) -> (b, a)
const swap = ([a, b]) => [b, a];
// Map each element of a structure to an action,
// evaluate these actions from left to right,
// and collect the results.
// traverse :: (a -> [b]) -> [a] -> [[b]]
const traverse = (f, xs) => {
const cons_f = (a, x) => ap(f(x)
.map(curry(cons)), a);
return xs.reduceRight(cons_f, [
[]
]);
};
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = (f, xs, ys) => {
const ny = ys.length;
return (xs.length <= ny ? xs : xs.slice(0, ny))
.map((x, i) => f(x, ys[i]));
};
// TEST ------------------------------------------------------------------
const matrixSize = 5;
return unlines(
zipWith(
(s, xs) => unlines(concat([
[s], xs.map(show), ['']
])), ["Lower", "Upper", "Symmetric"],
ap(
map(curry(pascalMatrix), [
id, // Lower
swap, // Upper
([a, b]) => [a + b, a] // Symmetric
]), [matrixSize]
)
)
);
})();
{{Out}}
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]
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:
[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
jq
{{works with|jq|1.4}}
# Generic functions
# Note: 'transpose' is defined in recent versions of jq
def transpose:
if (.[0] | length) == 0 then []
else [map(.[0])] + (map(.[1:]) | transpose)
end ;
# 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 ;
# 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)");
# 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 ;
# 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]) ) ;
'''Example''':
5
| ("\nUpper:", (pascal_upper | pp),
"\nLower:", (pascal_lower | pp),
"\nSymmetric:", (pascal_symmetric | pp)
)
{{out}}
$ 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
Kotlin
// 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))
}
{{out}}
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
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
{{out}}
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
Mathematica
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:
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"}}]]
{{out}}
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:
lowerPascal[size_] :=
MatrixExp[
SparseArray[{Band[{2, 1}] -> Range[size - 1]}, {size, size}]]]
But since the builtin function MatrixExp works by first computing eigenvalues this is likely to be slower for large Pascal matrices
PARI/GP
Pl(n)={matpascal(n-1)}
printf("%d",Pl(5))
{{out}}
[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]
Pu(n)={Pl(n)~}
printf("%d",Pu(5))
{{out}}
[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]
Ps(n)={matrix(n,n,n,g,binomial(n+g-2,n-1))}
printf("%d",Ps(5))
{{out}}
[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
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.
{{out}}
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
#!/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));
{{out}}
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
Perl 6
{{Works with|rakudo|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.
# 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;$j];
}
# West North NorthWest
@m[$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;
}
# 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 } }
# 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 '';
}
{{out}}
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
{{trans|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_StrFmt,-2,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))
{{out}}
### 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
(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)
{{out}}
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
{{trans|Rexx}}
*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;
{{out}}
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
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
{{out}}
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: Summing adjacent values
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))
{{out}}
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: Using a binomial coefficient generator function
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)]
{{out}} (As above)
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
}
# Alternate version
lower.pascal.alt <- function(n) {
a <- matrix(0, n, n)
a[, 1] <- 1
if (n > 1) {
for (j in 2:n) {
i <- j:n
a[i, j] <- cumsum(a[i - 1, j - 1])
}
}
a
}
# While it's possible to modify lower.pascal to get the upper matrix,
# here we simply transpose the lower one.
upper.pascal <- function(n) t(lower.pascal(n))
symm.pascal <- function(n) {
a <- matrix(0, n, n)
a[, 1] <- 1
for (i in 2:n) {
a[, i] <- cumsum(a[, i - 1])
}
a
}
The results follow
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
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)))
{{out}}
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
REXX
separate generation
Commentary: 1/3 of the REXX program deals with the displaying of the matrix.
/*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=='' 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
'''output''' 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).
/*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=='' 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
'''output''' is identical to the 1st REXX version.
Ring
# 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
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:'''
#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
{{out}}
[[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:
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)
{{out}}
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
//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
}
}
Sidef
{{trans|Perl 6}}
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)
{|i| m[i+1][s] = callback(m[i+1][s-1], m[i][s], m[i][s-1])} * (s-1)
{|i| m[s][i+1] = callback(m[s][i], m[s-1][i+1], m[s-1][i])} * (s)
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 = [[1]]
{ f(r) } * n
transpose(r)
}
}
functions.map { |f|
f(4).map { .map{ '%2s' % _ }.join(' ') }.join("\n")
}.join("\n\n").say
{{out}}
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.
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
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:
: 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 |
+--------------------------+
The last is a symmetric matrix, but Stata only shows the lower triangular part of symmetric matrices.
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
}
{{out}}
* 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
{{trans|Phix}}
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
{{out}}
### 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
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)
{{Out}}
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
zkl
{{trans|Python}}
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)
}
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(_));
{{out}}
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