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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dyck path ： ウィキペディア英語版 Catalan number In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894). Using zero-based numbering, the ''n''th Catalan number is given directly in terms of binomial coefficients by :$C\_n\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash prod\backslash limits\_^\backslash frac\; \backslash qquad\backslash mboxn\backslash ge\; 0.$ The first Catalan numbers 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, … . == Properties == An alternative expression for ''C''''n'' is :$C\_n\; =\; -\; =\; \backslash quad\backslash textn\backslash ge\; 0,$ which is equivalent to the expression given above because $\backslash tbinom=\backslash tfrac\backslash tbinomn$. This shows that ''C''''n'' 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. The Catalan numbers satisfy the recurrence relation :$C\_0\; =\; 1\; \backslash quad\; \backslash mbox\; \backslash quad\; C\_=\backslash sum\_^C\_i\backslash ,C\_\backslash quad\backslash textn\backslash ge\; 0;$ moreover, :$C\_n=\; \backslash frac\; 1\; \backslash sum\_^n\; ^2.$ This is because $\backslash tbinom\; =\; \backslash sum\_^n\; \backslash tbinom^2,$ since choosing ''n'' numbers from a 2''n'' set of numbers can be uniquely divided into 2 parts: choosing ''i'' numbers out of the first ''n'' numbers and then choosing ''n''-''i'' numbers from the remaining ''n'' numbers. They also satisfy: :$C\_0\; =\; 1\; \backslash quad\; \backslash mbox\; \backslash quad\; C\_=\backslash fracC\_n,$ which can be a more efficient way to calculate them. Asymptotically, the Catalan numbers grow as :$C\_n\; \backslash sim\; \backslash frac\}$ in the sense that the quotient of the ''n''th Catalan number and the expression on the right tends towards 1 as ''n'' → +∞. Some sources use just $C\_n\; \backslash sim\; \backslash frac\backslash sqrt\}.$ This means that the Catalan numbers are a solution of the Hausdorff moment problem on the interval (4 ) instead of (1 ). The orthogonal polynomials having the weight function $\backslash rho(x)$ on $(4)$ are :$H\_n(x)=\backslash sum\_^n(-x)^k.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Catalan number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.