Catalan Number Calculator
Find the n-th Catalan number and see how the sequence 1, 1, 2, 5, 14, 42, … explodes as n grows.
Counts balanced parentheses, full binary trees and polygon triangulations.
- 1
Seed value C₀ = 1
1 - 2
Recurrence ratio for step n
2 × (2 × 8 − 1) ÷ (8 + 1) = 3.333333Each Catalan number equals the previous term multiplied by 2(2n−1)/(n+1). - 3
Previous term C7
429 - 4
C8 = C7 × ratio
429 × 3.333333 = 1,430
How does this calculator work?
The n-th Catalan number is Cₙ = C(2n, n)/(n+1) = (2n)!/((n+1)!·n!). It counts balanced parentheses, full binary trees, lattice paths, and polygon triangulations. The sequence starts 1, 1, 2, 5, 14, 42, 132, … and grows about like 4ⁿ. Enter any integer n ≥ 0 to get Cₙ and the full growth curve.
Formula
How this is calculated
Enter a non-negative integer n. The Catalan numbers Cₙ count an enormous family of combinatorial objects: the number of ways to correctly match n pairs of parentheses, the number of full binary trees with n+1 leaves, the number of monotonic lattice paths that stay below the diagonal, and the number of triangulations of a convex (n+2)-gon — all equal to the same Cₙ.
The closed form is Cₙ = (2n)! / ((n+1)! · n!), equivalently the central binomial coefficient C(2n, n) divided by (n+1). Computing factorials directly overflows quickly, so this calculator uses the exact integer recurrence C₀ = 1 and C_{k+1} = C_k · 2(2k+1)/(k+2). Each step multiplies by an integer-valued factor, keeping the running value precise for as long as double-precision allows and building the full sequence C₀…Cₙ plotted in the growth curve.
The values are dimensionless counts, so only integer n ≥ 0 is meaningful; fractional input is floored. Because Cₙ grows roughly like 4ⁿ / (n^{3/2}√π), the numbers exceed the exact integer range of a 64-bit float beyond about n = 30 and overflow entirely past n ≈ 170, so the input is capped at 170 and large results are shown in scientific notation.
Frequently asked questions
The sequence begins C₀ = 1, C₁ = 1, C₂ = 2, C₃ = 5, C₄ = 14, C₅ = 42, C₆ = 132, C₇ = 429, and C₈ = 1430.
From the recurrence, Cₙ/Cₙ₋₁ = 2(2n−1)/(n+1), which tends to 4 as n grows. This reflects the asymptotic growth Cₙ ≈ 4ⁿ / (n^{3/2}√π).
Catalan numbers grow exponentially. Beyond n ≈ 170 the value exceeds the maximum representable double-precision number (~1.8×10³⁰⁸), so larger inputs cannot be computed exactly with standard floating point.
Also known as
TG we-Calculate Editorial Team. (2026). Catalan Number Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/catalan-number-calculator
TG we-Calculate Editorial Team. "Catalan Number Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/catalan-number-calculator.
TG we-Calculate Editorial Team, "Catalan Number Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/catalan-number-calculator
@misc{wecalculate_catalan_number_calculator, title = {Catalan Number Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/catalan-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
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