Catalan numbers
1, 1, 2, 5, 14, 42, 132, 429… The Catalan numbers turn up whenever you count balanced structures, and the surprise is how many unrelated-looking problems share the same answer.
Updated on October 7, 2026
The first Catalan numbers
Computed from the recurrence C(k+1) = C(k)·2(2k+1)/(k+2), which stays in whole numbers at every step. We use big integers because C(30) already passes what a double-precision float can hold exactly.
| n | C(n) |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 5 |
| 4 | 14 |
| 5 | 42 |
| 6 | 132 |
| 7 | 429 |
| 8 | 1,430 |
| 9 | 4,862 |
| 10 | 16,796 |
| 11 | 58,786 |
| 12 | 208,012 |
| 13 | 742,900 |
| 14 | 2,674,440 |
| 15 | 9,694,845 |
| 16 | 35,357,670 |
| 17 | 129,644,790 |
| 18 | 477,638,700 |
| 19 | 1,767,263,190 |
| 20 | 6,564,120,420 |
Source: Cálculo próprio a partir dos dados citados nesta página
Four problems with the same answer
C(n) counts: the ways to correctly match n pairs of brackets; the ways to cut a convex polygon with n+2 sides into triangles using non-crossing diagonals; the binary trees with n nodes; and the paths on a grid from one corner to the opposite one that never cross the diagonal. All four give 14 for n = 4, and the reason is that each can be turned into any other by a relabelling.
The closed form is C(n) = (2n choose n)/(n+1), which also explains the growth: C(n) is roughly 4ⁿ/(n^1.5·√π), so the numbers grow fast but not as fast as factorials. Eugène Catalan studied them in the 1830s, though Euler had already counted polygon triangulations a century earlier.
Source: OEIS — The On-Line Encyclopedia of Integer Sequences
How to read this page
How to read the numbers here
Every value on "Catalan numbers" comes from the source named beside its section, never from an estimate of our own with no origin.
Where the data comes from
Each section of "Catalan numbers" names its official source; the verification date sits at the top of the page.
Linked to the rest of the module
"Catalan numbers" is one of this module’s reference pages; the glossary below defines the technical terms used here.
Words on this page
Standard uncertainty
The margin attached to a measured value, usually written in parentheses on the last digits. A constant that is "exact by definition" has zero standard uncertainty; a measured one, like the gravitational constant G, always carries some.
Standard definition by BIPM — Bureau International des Poids et Mesures (SI) · Standard uncertainty
Mole
The SI counting unit for atoms, molecules or other entities: one mole is exactly 6.02214076 × 10²³ of them, by the definition fixed in 2019. It is what ties a mass in grams to a number of particles.
Standard definition by BIPM — Bureau International des Poids et Mesures (SI) · Mole
Atomic number
The number of protons in an atom’s nucleus, written Z. It is this number, not the mass, that defines which element an atom is: every atom with 8 protons is oxygen, whether it has 8 or 18 neutrons.
Standard definition by IUPAC · Atomic number
Atomic weight
The average weight of an element’s atoms, weighted by how abundant each isotope is in nature, in atomic mass units. For 14 elements CIAAW publishes a range instead of a single value, because that abundance varies a little from sample to sample.
Standard definition by CIAAW — Commission on Isotopic Abundances and Atomic Weights · Atomic weight