Catalan Number Calculator
Calculate Catalan numbers and explore their many combinatorial applications.
Calculate C_n
C_5
42
= (2×5)! / ((5+1)! × 5!) = 10! / (6! × 5!)
Catalan Sequence
C0=1C1=1C2=2C3=5C4=14C5=42C6=132C7=429C8=1430C9=4862C10=16796C11=58786
Combinatorial Interpretations of C_5 = 42
Balanced Parentheses
42 ways to arrange 5 pairs
Binary Trees
42 different full binary trees with 6 leaves
Triangulations
42 ways to triangulate a polygon with 7 sides
Dyck Paths
42 paths from (0,0) to (25,0)
Mountain Ranges
42 mountain ranges with 5 upstrokes
Formulas
Direct Formula
C_n = (2n)! / ((n+1)! × n!)
Binomial Form
C_n = C(2n,n) / (n+1)
Recurrence
C_n = Σ C_i × C_(n-1-i)
Asymptotic
C_n ~ 4^n / (n^(3/2) × √π)