The first 10 Catalan numbers:

Evaluate for large arguments:

Evaluate for half-integer arguments:

Evaluate numerically:

Evaluate for complex arguments:

Plot the Catalan number as a function of its index:

Compute sums involving CatalanNumber:

CatalanNumber threads element-wise over lists:

CatalanNumber can be used with Interval and CenteredInterval objects:

TraditionalForm typesetting:

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 a⊕b⊕c⊕d:

Check:

The Catalan numbers CatalanNumber[n] can be characterized as the unique set of numbers such that two Hankel determinants are both equal to one. Verify for the first few cases:

Verify an expression for the Catalan numbers in terms of double factorials:

The generating function for Catalan numbers:

Catalan numbers can be represented as a difference of binomial coefficients:

Catalan numbers can be represented in terms of the generalized Bell polynomial:

CatalanNumber can be represented as a DifferenceRoot:

FindSequenceFunction can recognize the CatalanNumber sequence:

The exponential generating function for CatalanNumber:

The Catalan number is, by convention, defined using its representation in terms of binomials:

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

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

Determinants of Hankel matrices made out of sums of Catalan numbers:

Compare with an expression in terms of the Fibonacci numbers: