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Catalan number - Properties of the Catalan numbers | | Catalan number - Properties of the Catalan numbers: Encyclopedia II - Catalan number - Properties of the Catalan numbers | | 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 | | | Catalan number, Catalan number - Applications in combinatorics, Catalan number - First proof: using generating functions, Catalan number - Hankel matrix, Catalan number - History, Catalan number - Proof of the formula, Catalan number - Properties of the Catalan numbers, Catalan number - Second proof, Catalan number - Third proof | | |
| | | Catalan number: Encyclopedia II - Catalan number - Properties of the Catalan numbers
Catalan number - Properties of the Catalan numbers
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, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452...
Asymptotically, the Catalan numbers grow as
in the sense that the quotient of the n-th Catalan number and the expression on the right tends towards 1 for n → ∞. (This can be proved by using Stirling's approximation for n!.)
Other related archives1, 132, 14, 18th century, 2, 42, 5, Belgian, Bell number, Dyck language, Eugène Charles Catalan, Hankel matrix, Leonhard Euler, OEIS, Richard Stanley, Stirling's approximation, Wigner semicircle law, bijective proofs, binomial coefficients, binomial theorem, combinatorial mathematics, combinatorics, complex analysis, convex polygon, cumulants, determinant, free probability, generating function, mathematical induction, mathematician, natural number, natural numbers, noncrossing partitions, parenthesis, parenthesized, permutations, random matrices, recurrence relation, recursive, sequence, straight lines, string, tends towards, trees, triangles
 Adapted from the Wikipedia article "Properties of the Catalan numbers", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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