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Catalan number
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Catalan number
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu.
The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
The first Catalan numbers for n = 0, 1, 2, 3, ... are
An alternative expression for Cn is
which is equivalent to the expression given above because . This expression shows that Cn is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula.
Another alternative expression is
which can be directly interpreted in terms of the cycle lemma; see below.
The Catalan numbers satisfy the recurrence relations
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Catalan number
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu.
The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
The first Catalan numbers for n = 0, 1, 2, 3, ... are
An alternative expression for Cn is
which is equivalent to the expression given above because . This expression shows that Cn is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula.
Another alternative expression is
which can be directly interpreted in terms of the cycle lemma; see below.
The Catalan numbers satisfy the recurrence relations
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