Fuss–Catalan Numbers are a generalization of Catalan numbers that uses triplets instead of pairs.
The Fuss-Catalan Numbers can be represented by a Series with the formula:
The first few Fuss–Catalan Numbers are
1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675………..
for n = 0, 1, 2, 3, … respectively
Applications of Fuss-Catalan number:
-
Count the number of ways to place parentheses among of 2n+1 numbers to be grouped three at a time.
Example: There are 3 ways to parenthesize {1, 2, 3, 4, 5} as triplets:
{{1, 2, 3}, 4, 5}, {1, {2, 3, 4}, 5}, {1, 2, {3, 4, 5}}
- Count the number of complete ternary trees with n internal nodes.
- Count the number of paths of length 3n through a 2n-by-n grid that does not cross above the main diagonal
Example: There are 3 paths from (0, 0) to (4, 2) that don’t cross above the diagonal:
- and many more. Please refer this link for more applications
Implementation of Fuss-Catalan number:
C++
// C++ program for nth Fuss–Catalan Number #include <iostream> using namespace std;
// Returns value of Binomial Coefficient C(n, k) unsigned long int binomialCoeff(unsigned int n,
unsigned int k)
{ unsigned long int res = 1;
// Since C(n, k) = C(n, n-k)
if (k > n - k)
k = n - k;
// Calculate value of
//[n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
for ( int i = 0; i < k; ++i) {
res *= (n - i);
res /= (i + 1);
}
return res;
} // A Binomial coefficient based function // to find nth Fuss–Catalan number in O(n) time unsigned long int Fuss_catalan(unsigned int n)
{ // Calculate value of 3nCn
unsigned long int c = binomialCoeff(3 * n, n);
// return 3nCn/(2n+1)
return c / (2 * n + 1);
} // Driver code int main()
{ for ( int i = 0; i < 10; i++)
cout << Fuss_catalan(i) << " " ;
return 0;
} |
Java
// Java program for nth Fuss-Catalan Number class GFG
{ // Returns value of Binomial Coefficient C(n, k) static int binomialCoeff( int n, int k)
{ int res = 1 ;
// Since C(n, k) = C(n, n-k)
if (k > n - k)
k = n - k;
// Calculate value of
//[n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
for ( int i = 0 ; i < k; ++i)
{
res *= (n - i);
res /= (i + 1 );
}
return res;
} // A Binomial coefficient based function // to find nth Fuss-Catalan number in O(n) time static int Fuss_catalan( int n)
{ // Calculate value of 3nCn
int c = binomialCoeff( 3 * n, n);
// return 3nCn/(2n+1)
return c / ( 2 * n + 1 );
} // Driver code public static void main(String []args)
{ for ( int i = 0 ; i < 10 ; i++)
System.out.print(Fuss_catalan(i) + " " );
} } // This code is contributed by 29AjayKumar |
Python3
# Python3 program for nth Fuss–Catalan Number # Returns value of Binomial Coefficient C(n, k) def binomialCoeff(n, k) :
res = 1 ;
# Since C(n, k) = C(n, n-k)
if (k > n - k) :
k = n - k;
# Calculate value of
# [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
for i in range (k) :
res * = (n - i);
res / / = (i + 1 );
return res;
# A Binomial coefficient based function # to find nth Fuss–Catalan number in O(n) time def Fuss_catalan(n) :
# Calculate value of 3nCn
c = binomialCoeff( 3 * n, n);
# return 3nCn/(2n+1)
return c / / ( 2 * n + 1 );
# Driver code if __name__ = = "__main__" :
for i in range ( 10 ) :
print (Fuss_catalan(i), end = " " );
# This code is contributed by AnkitRai01 |
C#
// C# program for nth Fuss-Catalan Number using System;
class GFG
{ // Returns value of Binomial Coefficient C(n, k) static int binomialCoeff( int n, int k)
{ int res = 1;
// Since C(n, k) = C(n, n-k)
if (k > n - k)
k = n - k;
// Calculate value of
//[n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
for ( int i = 0; i < k; ++i)
{
res *= (n - i);
res /= (i + 1);
}
return res;
} // A Binomial coefficient based function // to find nth Fuss-Catalan number in O(n) time static int Fuss_catalan( int n)
{ // Calculate value of 3nCn
int c = binomialCoeff(3 * n, n);
// return 3nCn/(2n+1)
return c / (2 * n + 1);
} // Driver code public static void Main(String []args)
{ for ( int i = 0; i < 10; i++)
Console.Write(Fuss_catalan(i) + " " );
} } // This code is contributed by PrinciRaj1992 |
Javascript
<script> // Javascript program for nth Fuss–Catalan Number // Returns value of Binomial Coefficient C(n, k) function binomialCoeff(n, k)
{ var res = 1;
// Since C(n, k) = C(n, n-k)
if (k > n - k)
k = n - k;
// Calculate value of
//[n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
for ( var i = 0; i < k; ++i) {
res *= (n - i);
res = parseInt(res / (i + 1));
}
return res;
} // A Binomial coefficient based function // to find nth Fuss–Catalan number in O(n) time function Fuss_catalan(n)
{ // Calculate value of 3nCn
var c = binomialCoeff(3 * n, n);
// return 3nCn/(2n+1)
return parseInt(c / (2 * n + 1));
} // Driver code for ( var i = 0; i < 10; i++)
document.write(Fuss_catalan(i)+ " " );
</script> |
Output:
1 1 3 12 55 273 1428 7752 43263 246675
Time Complexity: O(n)
Auxiliary Space: O(1)