In mathematics, particularly in matrix theory and combinatorics, the Pascal Matrix is an infinite matrix containing binomial coefficients as its elements. There are three ways to achieve this: as either an upper-triangular matrix, a lower-triangular matrix, or a symmetric matrix.
The 5 x 5 truncations of these are shown below:
The elements of the symmetric Pascal Matrix are the binomial coefficient, i.e
Given a positive integer n. The task is to print the Symmetric Pascal Matrix of size n x n.
Examples:
Input : n = 5 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
Below is the code to implement n x n symmetric pascal matrix:
// CPP Program to print symmetric pascal matrix. #include <bits/stdc++.h> using namespace std;
// Print Pascal Matrix void printpascalmatrix( int n)
{ int C[2 * n + 1][2 * n + 1] = { 0 };
// Calculate value of Binomial Coefficient in
// bottom up manner
for ( int i = 0; i <= 2 * n; i++) {
for ( int j = 0; j <= min(i, 2 * n); j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using previously
// stored values
else
C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
}
}
// Printing the pascal matrix
for ( int i = 0; i < n; i++) {
for ( int j = 0; j < n; j++)
cout << C[i + j][i] << " " ;
cout << endl;
}
} // Driven Program int main()
{ int n = 5;
printpascalmatrix(n);
return 0;
} |
// java Program to print // symmetric pascal matrix. import java.io.*;
class GFG
{ // Print Pascal Matrix
static void printpascalmatrix( int n)
{
int C[][] = new int [ 2 * n + 1 ][ 2 * n + 1 ];
// Calculate value of Binomial Coefficient in
// bottom up manner
for ( int i = 0 ; i <= 2 * n; i++)
{
for ( int j = 0 ; j <= Math.min(i, 2 * n); j++)
{
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1 ;
// Calculate value using previously
// stored values
else
C[i][j] = C[i - 1 ][j - 1 ]
+ C[i - 1 ][j];
}
}
// Printing the pascal matrix
for ( int i = 0 ; i < n; i++)
{
for ( int j = 0 ; j < n; j++)
System.out.print ( C[i + j][i] + " " );
System.out.println();
}
}
// Driven Program
public static void main (String[] args)
{
int n = 5 ;
printpascalmatrix(n);
}
} // This code is contributed by vt_m. |
# Python3 Program to print # symmetric pascal matrix. # Print Pascal Matrix def printpascalmatrix(n):
C = [[ 0 for x in range ( 2 * n + 1 )]
for y in range ( 2 * n + 1 )]
# Calculate value of
# Binomial Coefficient
# in bottom up manner
for i in range ( 2 * n + 1 ):
for j in range ( min (i, 2 * n) + 1 ):
# Base Cases
if (j = = 0 or j = = i):
C[i][j] = 1 ;
# Calculate value
# using previously
# stored values
else :
C[i][j] = (C[i - 1 ][j - 1 ] +
C[i - 1 ][j]);
# Printing the
# pascal matrix
for i in range (n):
for j in range (n):
print (C[i + j][i],
end = " " );
print ();
# Driver Code n = 5 ;
printpascalmatrix(n); # This code is contributed by mits |
// C# program to print // symmetric pascal matrix. using System;
class GFG {
// Print Pascal Matrix
static void printpascalmatrix( int n)
{
int [, ] C = new int [2 * n + 1, 2 * n + 1];
// Calculate value of Binomial Coefficient
// in bottom up manner
for ( int i = 0; i <= 2 * n; i++) {
for ( int j = 0; j <= Math.Min(i, 2 * n); j++) {
// Base Cases
if (j == 0 || j == i)
C[i, j] = 1;
// Calculate value using previously
// stored values
else
C[i, j] = C[i - 1, j - 1]
+ C[i - 1, j];
}
}
// Printing the pascal matrix
for ( int i = 0; i < n; i++) {
for ( int j = 0; j < n; j++)
Console.Write(C[i + j, i] + " " );
Console.WriteLine();
}
}
// Driven Program
public static void Main()
{
int n = 5;
printpascalmatrix(n);
}
} // This code is contributed by vt_m. |
<?php // PHP Program to print symmetric // pascal matrix. // Print Pascal Matrix function printpascalmatrix( $n )
{ $C [2 * $n + 1][2 * $n + 1] = (0);
// Calculate value of Binomial
// Coefficient in bottom up manner
for ( $i = 0; $i <= 2 * $n ; $i ++)
{
for ( $j = 0; $j <= min( $i , 2 * $n ); $j ++)
{
// Base Cases
if ( $j == 0 || $j == $i )
$C [ $i ][ $j ] = 1;
// Calculate value
// using previously
// stored values
else
$C [ $i ][ $j ] = $C [ $i - 1][ $j - 1] +
$C [ $i - 1][ $j ];
}
}
// Printing the pascal matrix
for ( $i = 0; $i < $n ; $i ++) {
for ( $j = 0; $j < $n ; $j ++)
echo $C [ $i + $j ][ $i ], " " ;
echo "\n" ;
}
} // Driver Code
$n = 5;
printpascalmatrix( $n );
// This code is contributed by aj_36 ?> |
<script> // JavaScript Program to print // symmetric pascal matrix. // Print Pascal Matrix
function printpascalmatrix(n)
{
let C = new Array(2 * n + 1);
// Loop to create 2D array using 1D array
for ( var i = 0; i < C.length; i++) {
C[i] = new Array(2);
}
// Calculate value of Binomial Coefficient in
// bottom up manner
for (let i = 0; i <= 2 * n; i++)
{
for (let j = 0; j <= Math.min(i, 2 * n); j++)
{
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using previously
// stored values
else
C[i][j] = C[i - 1][j - 1]
+ C[i - 1][j];
}
}
// Printing the pascal matrix
for (let i = 0; i < n; i++)
{
for (let j = 0; j < n; j++)
document.write( C[i + j][i] + " " );
document.write( "<br/>" );
}
}
// Driver code let n = 5;
printpascalmatrix(n);
</script> |
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
Time Complexity: O(N2)
Auxiliary Space: O(N2)