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mathematical analysis > combinatorics > binomial coefficient > Catalan number

algebra > combinatorics > binomial coefficient > Catalan number

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Catalan number

In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan. The nth Catalan number can be expressed directly in terms of the central binomial coefficients by
${\\displaystyle C_{n}={\ rac {1}{n+1}}{2n \\choose n}={\ rac {(2n)!}{(n+1)!\\,n!}}=\\prod \\limits _{k=2}^{n}{\ rac {n+k}{k}}\\qquad {\ ext{for }}n\\geq 0.}$
The first Catalan numbers for n = 0, 1, 2, 3, ... are
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... (sequence A000108 in the OEIS).

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Catalan_number)

binomial coefficient

nombre de Catalan

French

URI

http://data.loterre.fr/ark:/67375/PSR-N05ZGM7Q-N

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