Catalan Numbers Calculator
Catalan number with formula and meaning.
What it does
Catalan numbers count shapes: parenthesizations, binary trees, and mountain paths all total the same sequence 1, 1, 2, 5, 14. One formula rules every reading.
The tool above takes index n. It returns C(n) with the formula and meaning.
The default n = 10 gives 16796 from 184756 over 11. The n = 4 case gives 14, and n = 0 gives 1.
Sequence references tabulate Catalan from the binomial quotient with matching openers. Combinatorics texts count the same trees and paths beside.
Shapes multiply fast: C(10) already nears seventeen thousand while C(20) passes billions. Exact integer math holds this page through 25.
How to use
- Type n from 0 to 25.
- Read C(n) first.
- Read the formula beside it.
- Read the meaning last.
- Compare small cases by hand.
- Respect the 25 cap.
Formula
- n
- Index 0 to 25.
- C(n)
- nth Catalan number.
OEIS tabulates Catalan numbers with the binomial quotient definition. LibreTexts combinatorics mirrors the same parenthesization and tree counting. Multiplicative stepping keeps integers exact through n = 25.
Worked example
Example: Catalan 10
Default: n = 10.
- Index: n = 10.
- Binomial: C(20,10) = 184756.
- Divide: 184756 / 11 = 16796.
- Check: OEIS matches 16796.
- Meaning: binary trees and paths count.
- Tool shows C(10) = 16796, (2n)!/((n+1)!n!), Counts binary trees and paths.
Reference table
| n | C(n) | Note |
|---|---|---|
| 0 | 1 | Seed |
| 1 | 1 | Seed |
| 4 | 14 | 3 + paths |
| 10 | 16796 | Trees |
| 15 | 9694845 | Large |
Common mistakes
Try it yourself
Quick checks. Answers sit tucked under each question.
n = 4?
C(4) = 14 from 70/5.
n = 0?
C(0) = 1, the empty product seed.
n = 99?
Past 25 floats round: enter 0 to 25.
Questions
Catalan formula?
(2n)! over (n+1)!n!. C(4) = 70/5 = 14.
Openers?
1, 1, 2, 5, 14, 42, 132, 429, 1430, 16796 at n = 0 through 10.
Counts what?
Parenthesizations, binary trees, Dyck paths, triangulations all count Catalan.
Exact values?
Yes, integer stepping stays exact through n = 25 here.
Whole sequence?
No, this runs one index. OEIS tables list ranges.
Fibonacci same?
No, cousins with other recurrences need their own pages.