gives the n^(th) Catalan number .


  • CatalanNumber[n] is generically defined as .
  • Catalan numbers are integers for integer arguments, and appear in various tree enumeration problems.


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Basic Examples  (1)

The first 10 Catalan numbers:

Scope  (7)

Evaluate for large arguments:

Evaluate for half-integer arguments:

Evaluate numerically:

Evaluate for complex arguments:

Compute sums involving CatalanNumber:

CatalanNumber threads element-wise over lists:

TraditionalForm typesetting:

Generalizations & Extensions  (1)

Plot the Catalan number as a function of its index:

Applications  (1)

Compute the number of different ways to parenthesize an expression:

Distribute over lists in CirclePlus:

Use the pattern matcher to repeatedly split the list into two parts in all possible ways:

The number of ways to parenthesize the expression abcd:


Properties & Relations  (5)

The generating function for Catalan numbers:

Catalan numbers can be represented as difference of binomial coefficients:

CatalanNumber can be represented as a DifferenceRoot:

FindSequenceFunction can recognize the CatalanNumber sequence:

The exponential generating function for CatalanNumber:

Possible Issues  (1)

Catalan number TemplateBox[{{-, 1}}, CatalanNumber] is, by convention, defined using its representation in terms of binomials:

This value is different from the limiting value of the analytic function:

Neat Examples  (2)

The only odd Catalan numbers are those of the form CatalanNumber[2k-1]:

Determinants of Hankel matrices made out of Catalan numbers:

Compare to Fibonacci numbers:

Introduced in 2007
Updated in 2014