In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems; which often have a recursive flavour. They are named for the Belgian mathematician Eugène Charles Catalan (1814–1894). The nth Catalan number is given directly in terms of binomial coefficients by Catalan number - Properties of the Catalan numbers. One can verify that an alternative expression for Cn is ...

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• Catalan number - Properties of the Catalan numbers
• Catalan number - Applications in combinatorics
• Catalan number - Proof of the formula
• Catalan number - First proof: using generating functions
• Catalan number - Second proof
• Catalan number - Third proof
• Catalan number - Hankel matrix
• Catalan number - History

Read more here: » Catalan number: Encyclopedia - Catalan number

There are several ways of explaining why the formula given for Cn is correct; that is, why it solves the combinatorial problems listed above. The first proof below uses a generating function, and is not particularly illuminating. The second and third proofs are examples of bijective proofs; they involve literally counting a collection of some kind of object to arrive at the correct formula. Catalan number - First proof: using generating functions. The Catalan numbe ...

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Catalan number, Catalan number - Properties of the Catalan numbers, Catalan number - Applications in combinatorics, Catalan number - Proof of the formula, Catalan number - First proof: using generating functions, Catalan number - Second proof, Catalan number - Third proof, Catalan number - Hankel matrix, Catalan number - History

Read more here: » Catalan number: Encyclopedia II - Catalan number - Proof of the formula

One can verify that an alternative expression for Cn is This shows that Cn is a natural number, which is not a priori obvious from the first formula given. This expression forms the basis for André's proof of the correctness of the formula (see below under second proof). The first Catalan numbers (sequence A000108 in OEIS) for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 2446626702 ...

See also:

Catalan number, Catalan number - Properties of the Catalan numbers, Catalan number - Applications in combinatorics, Catalan number - Proof of the formula, Catalan number - First proof: using generating functions, Catalan number - Second proof, Catalan number - Third proof, Catalan number - Hankel matrix, Catalan number - History

Read more here: » Catalan number: Encyclopedia II - Catalan number - Properties of the Catalan numbers