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Fibonomial coefficient and Fibonomial triangle

  • Difficulty Level : Easy
  • Last Updated : 08 Jun, 2021

Fibonomial Coefficient 
In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as
\binom{n}{k}_f = \frac{F_n F_{n-1}....F_{n-k+1} }{F_k F_{k-1}....F1} = \frac{n_F!}{k_F!(n-k)_F!}
where n and k are non-negative integers, 0 ≤ k ≤ n, Fj is the j-th Fibonacci number and n!F is the nth Fibonorial, where 0!F, being the empty product, evaluates to 1. 
The Fibonomial coefficients are all integers. Some special values are: 
\binom{n}{0}_f = \binom{n}{n}_f = 1
\binom{n}{1}_f = \binom{n}{n-1}_f = F_n
\binom{n}{2}_f = \binom{n}{n-2}_f = \frac{F_nF_{n-1}}{F_2F_1}
\binom{n}{3}_f = \binom{n}{n-3}_f = \frac{F_nF_{n-1}F_{n-2}}{F_3F_2F_1}
\binom{n}{k}_f = \binom{n}{n-k}_f
Fibonomial triangle 
The Fibonomial coefficients are similar to binomial coefficients and can be displayed in a triangle similar to Pascal’s triangle. The first eight rows are shown below. 
 

The Recurrence Relation for Fibonomial Triangle: 
\binom{n}{k}_f = F_{n-k+1} \binom{n-1}{k-1}_f + F_{k-1} \binom{n-1}{k}_f
Given a positive integer n. The task is to print Fibonomial triangle of height n (or n + 1 rows).
Examples: 
 

Input : n = 6
Output :
1
1 1
1 1 1
1 2 2 1
1 3 6 3 1
1 5 15 15 5 1
1 8 40 60 40 8 1

Input : n = 5
Output :
1
1 1
1 1 1
1 2 2 1
1 3 6 3 1
1 5 15 15 5 1

 

Below is the implementation of printing Fibonomial Triangle of height n:
 



C++




// CPP Program to print Fibonomial Triangle of height n.
#include<bits/stdc++.h>
#define N 6
using namespace std;
 
// Function to produce Fibonacci Series.
void fib(int f[], int n)
{
    int i;
  
    /* 0th and 1st number of the series are 0 and 1*/
    f[0] = 0;
    f[1] = 1;
  
    for (i = 2; i <= n; i++)
     
        /* Add the previous 2 numbers in the series
         and store it */
        f[i] = f[i-1] + f[i-2];   
}
 
// Function to produce fibonomial coefficient
void fibcoef(int fc[][N+1], int f[], int n)
{
    for (int i = 0; i <= n; i++)
        fc[i][0] = 1;
         
    for (int i = 1; i <= n; i++)
    {
        for (int j = 1; j <= i; j++)
        {
            int k = j;
            while(k--)
                fc[i][j] *= f[k];
                 
            k = 1;
             
            while((j+1)!=k)
                fc[i][j] /= f[k++];
        }
    }
}
 
// Function to print Fibonomial Triangle.
void printFibonomialTriangle(int n)
{
    int f[N+1] = { 0 };
     
    // Finding the fibonacci series.
    fib(f, n);
     
    // to store triangle value.
    int dp[N+1][N+1] = { 0 };
     
    // initialising the 0th element of each row
    // and diagonal element equal to 0.
    for (int i = 0; i <= n; i++)
        dp[i][0] = dp[i][i] = 1;
         
    // for each row.
    for (int i = 1; i <= n; i++)
    {
        // for each column.
        for (int j = 1; j < i; j++)
         
            // finding each element using recurrence
            // relation.
            dp[i][j] = f[i-j+1]*dp[i-1][j-1] +
                       f[j-1]*dp[i-1][j];   
    }
     
    // printing the Fibonomial Triangle.
    for (int i = 0; i <= n; i++)
    {
        for (int j = 0; j <= i; j++)       
            cout << dp[i][j] << " ";       
        cout << endl;
    }
}
// Driven Program
int main()
{
    int n = 6;   
    printFibonomialTriangle(n);
    return 0;
}

Java




// Java Program to print Fibonomial
// Triangle of height n.
class GFG
{
    static final int N=6;
     
    // Function to produce Fibonacci Series.
    static void fib(int f[], int n)
    {
        int i;
     
        /* 0th and 1st number of
         the series are 0 and 1*/
        f[0] = 0;
        f[1] = 1;
     
        for (i = 2; i <= n; i++)
         
            /* Add the previous 2 numbers in
            the series and store it */
            f[i] = f[i-1] + f[i-2];
    }
     
    // Function to produce fibonomial coefficient
    static void fibcoef(int fc[][], int f[], int n)
    {
        for (int i = 0; i <= n; i++)
            fc[i][0] = 1;
             
        for (int i = 1; i <= n; i++)
        {
            for (int j = 1; j <= i; j++)
            {
                int k = j;
                 
                while(k > 0)
                {
                    k--;
                    fc[i][j] *= f[k];
                }
                     
                k = 1;
                 
                while((j + 1) != k)
                    fc[i][j] /= f[k++];
            }
        }
    }
     
    // Function to print Fibonomial Triangle.
    static void printFibonomialTriangle(int n)
    {
        int f[] = new int[N+1];
         
        // Finding the fibonacci series.
        fib(f, n);
         
        // to store triangle value.
        int dp[][] = new int[N + 1][N + 1];
         
        // initialising the 0th element of each row
        // and diagonal element equal to 0.
        for (int i = 0; i <= n; i++)
            dp[i][0] = dp[i][i] = 1;
             
        // for each row.
        for (int i = 1; i <= n; i++)
        {
            // for each column.
            for (int j = 1; j < i; j++)
             
                // finding each element using recurrence
                // relation.
                dp[i][j] = f[i - j + 1] * dp[i - 1][j - 1] +
                                           f[j-1]*dp[i-1][j];
        }
         
        // printing the Fibonomial Triangle.
        for (int i = 0; i <= n; i++)
        {
            for (int j = 0; j <= i; j++)    
                System.out.print(dp[i][j] + " ");    
            System.out.println();
        }
}
 
    // Driver code
    public static void main (String[] args)
    {
        int n = 6;
         
        printFibonomialTriangle(n);
    }
}
 
// This code is contributed by Anant Agarwal.

Python3




# Python3 Program to print Fibonomial
# Triangle of height n.
N = 6;
 
# Function to produce Fibonacci Series.
def fib(f, n):
 
    # 0th and 1st number of the
    # series are 0 and 1
    f[0] = 0;
    f[1] = 1;
 
    for i in range(2, n + 1):
     
        # Add the previous 2 numbers in
        # the series and store it
        f[i] = f[i - 1] + f[i - 2];
 
# Function to produce fibonomial
# coefficient
def fibcoef(fc, f, n):
 
    for i in range(n + 1):
        fc[i][0] = 1;
         
    for i in range(1, n + 1):
        for j in range(1, i + 1):
            k = j;
            while(k > 0):
                k -= 1;
                fc[i][j] *= f[k];
                 
            k = 1;
             
            while((j + 1) != k):
                fc[i][j] /= f[k];
                k += 1;
 
# Function to print Fibonomial Triangle.
def printFibonomialTriangle(n):
 
    f = [0] * (N + 1);
     
    # Finding the fibonacci series.
    fib(f, n);
     
    # to store triangle value.
    dp = [[0 for x in range(N + 1)]
             for y in range(N + 1)];
     
    # initialising the 0th element of each
    # row and diagonal element equal to 0.
    for i in range(n + 1):
        dp[i][0] = 1;
        dp[i][i] = 1;
         
    # for each row.
    for i in range(1, n + 1):
        # for each column.
        for j in range(1, i):
         
            # finding each element using
            # recurrence relation.
            dp[i][j] = (f[i - j + 1] * dp[i - 1][j - 1] +
                        f[j - 1] * dp[i - 1][j]);
     
    # printing the Fibonomial Triangle.
    for i in range(n + 1):
        for j in range(i + 1):    
            print(dp[i][j], end = " ");    
        print("");
     
# Driver Code
n = 6;
printFibonomialTriangle(n);
 
# This code is contributed by mits

C#




// C# Program to print Fibonomial
// Triangle of height n.
using System;
 
class GFG
{
    static int N = 6;
     
    // Function to produce Fibonacci Series.
    static void fib(int []f, int n)
    {
        int i;
     
        /* 0th and 1st number of
        the series are 0 and 1*/
        f[0] = 0;
        f[1] = 1;
     
        for (i = 2; i <= n; i++)
         
            /* Add the previous 2 numbers in
            the series and store it */
            f[i] = f[i - 1] + f[i - 2];
    }
     
    // Function to produce fibonomial coefficient
    static void fibcoef(int [,]fc, int []f, int n)
    {
        for (int i = 0; i <= n; i++)
            fc[i,0] = 1;
             
        for (int i = 1; i <= n; i++)
        {
            for (int j = 1; j <= i; j++)
            {
                int k = j;
                 
                while(k > 0)
                {
                    k--;
                    fc[i, j] *= f[k];
                }
                     
                k = 1;
                 
                while((j + 1) != k)
                    fc[i, j] /= f[k++];
            }
        }
    }
     
    // Function to print Fibonomial Triangle.
    static void printFibonomialTriangle(int n)
    {
        int []f = new int[N + 1];
         
        // Finding the fibonacci series.
        fib(f, n);
         
        // to store triangle value.
        int [,]dp = new int[N + 1, N + 1];
         
        // initialising the 0th element of each row
        // and diagonal element equal to 0.
        for (int i = 0; i <= n; i++)
            dp[i, 0] = dp[i, i] = 1;
             
        // for each row.
        for (int i = 1; i <= n; i++)
        {
            // for each column.
            for (int j = 1; j < i; j++)
             
                // finding each element using recurrence
                // relation.
                dp[i,j] = f[i - j + 1] * dp[i - 1,j - 1] +
                                    f[j - 1] * dp[i - 1, j];
        }
         
        // printing the Fibonomial Triangle.
        for (int i = 0; i <= n; i++)
        {
            for (int j = 0; j <= i; j++)
            Console.Write(dp[i,j] + " ");
            Console.WriteLine();
        }
}
 
    // Driver code
    public static void Main ()
    {
        int n = 6;
         
        printFibonomialTriangle(n);
    }
}
 
// This code is contributed by Vt_m.

PHP




<?php
// PHP Program to print Fibonomial Triangle of height n.
 
$N=6;
 
// Function to produce Fibonacci Series.
function fib(&$f, $n)
{
 
    /* 0th and 1st number of the series are 0 and 1*/
    $f[0] = 0;
    $f[1] = 1;
 
    for ($i = 2; $i <= $n; $i++)
     
        /* Add the previous 2 numbers in the series
        and store it */
        $f[$i] = $f[$i-1] + $f[$i-2];
}
 
// Function to produce fibonomial coefficient
function fibcoef($fc, $f, $n)
{
    for ($i = 0; $i <= $n; $i++)
        $fc[$i][0] = 1;
         
    for ($i = 1; $i <= $n; $i++)
    {
        for ($j = 1; $j <= $i; $j++)
        {
            $k = $j;
            while($k--)
                $fc[$i][$j] *= $f[$k];
                 
            $k = 1;
             
            while(($j+1)!=$k)
                $fc[$i][$j] /= $f[$k++];
        }
    }
}
 
// Function to print Fibonomial Triangle.
function printFibonomialTriangle($n)
{
    global $N;
    $f=array_fill(0,$N+1,0);
     
    // Finding the fibonacci series.
    fib($f, $n);
     
    // to store triangle value.
    $dp=array_fill(0,$N+1,array_fill(0,$N+1,0));
     
    // initialising the 0th element of each row
    // and diagonal element equal to 0.
    for ($i = 0; $i <= $n; $i++)
        $dp[$i][0] = $dp[$i][$i] = 1;
         
    // for each row.
    for ($i = 1; $i <= $n; $i++)
    {
        // for each column.
        for ($j = 1; $j < $i; $j++)
         
            // finding each element using recurrence
            // relation.
            $dp[$i][$j] = $f[$i-$j+1]*$dp[$i-1][$j-1] +
                    $f[$j-1]*$dp[$i-1][$j];
    }
     
    // printing the Fibonomial Triangle.
    for ($i = 0; $i <= $n; $i++)
    {
        for ($j = 0; $j <= $i; $j++)    
            echo $dp[$i][$j]." ";    
        echo "\n";
    }
}
// Driven Program
 
    $n = 6;
    printFibonomialTriangle($n);
 
// This code is contributed by mits
?>

Javascript




<script>
 
// Javascript Program to print Fibonomial
// Triangle of height n.
 
     var N = 6;
 
    // Function to produce Fibonacci Series.
    function fib(f , n) {
        var i;
 
        /*
         * 0th and 1st number of the series are 0 and 1
         */
        f[0] = 0;
        f[1] = 1;
 
        for (i = 2; i <= n; i++)
 
            /*
             * Add the previous 2 numbers in
             the series and store it
             */
            f[i] = f[i - 1] + f[i - 2];
    }
 
    // Function to produce fibonomial coefficient
    function fibcoef(fc , f , n) {
        for (i = 0; i <= n; i++)
            fc[i][0] = 1;
 
        for (i = 1; i <= n; i++) {
            for (j = 1; j <= i; j++) {
                var k = j;
 
                while (k > 0) {
                    k--;
                    fc[i][j] *= f[k];
                }
 
                k = 1;
 
                while ((j + 1) != k)
                    fc[i][j] /= f[k++];
            }
        }
    }
 
    // Function to prvar Fibonomial Triangle.
    function printFibonomialTriangle(n) {
        var f = Array(N + 1).fill(0);
 
        // Finding the fibonacci series.
        fib(f, n);
 
        // to store triangle value.
        var dp = Array(N + 1);
        for(var i =0;i<N+1;i++)
        dp[i] = Array(N + 1).fill(0);
 
        // initialising the 0th element of each row
        // and diagonal element equal to 0.
        for (i = 0; i <= n; i++)
            dp[i][0] = dp[i][i] = 1;
 
        // for each row.
        for (i = 1; i <= n; i++) {
            // for each column.
            for (j = 1; j < i; j++)
 
                // finding each element using recurrence
                // relation.
                dp[i][j] = f[i - j + 1] * dp[i - 1][j - 1] +
                f[j - 1] * dp[i - 1][j];
        }
 
        // printing the Fibonomial Triangle.
        for (i = 0; i <= n; i++) {
            for (j = 0; j <= i; j++)
                document.write(dp[i][j] + " ");
            document.write("<br/>");
        }
    }
 
    // Driver code
     
        var n = 6;
 
        printFibonomialTriangle(n);
 
// This code contributed by Rajput-Ji
 
</script>

Output: 
 

1 
1 1 
1 1 1 
1 2 2 1 
1 3 6 3 1 
1 5 15 15 5 1 
1 8 40 60 40 8 1 

 

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