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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 Double factorial ： ウィキペディア英語版 Double factorial In mathematics, the product of all the integers from 1 up to some non-negative integer ''n'' that have the same parity as ''n'' is called the double factorial or semifactorial of ''n'' and is denoted by ''n''!!.〔.〕 That is, :$n!!\; =\; \backslash prod\_^m\; (n-2k)\; =\; n\; (n-2)\; (n-4)\; \backslash cdots$ where $m=\backslash lceil\; n/2\; \backslash rceil\; -\; 1.$ A consequence of this definition is that (as an empty product) :$0!!\; =\; 1.$ For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945. For even ''n'' the double factorial is :$n!!\; =\; \backslash prod\_^\; (2k)\; =\; n(n-2)\backslash cdots\; 2.$ For odd ''n'' it is :$n!!\; =\; \backslash prod\_^\; (2k-1)\; =\; n(n-2)\backslash cdots\; 1.$ The sequence of double factorials for even ''n'' = 0, 2, 4, 6, 8, ... starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120, .... The sequence of double factorials for odd ''n'' = 1, 3, 5, 7, ... starts as : 1, 3, 15, 105, 945, 10395, 135135, .... (possibly the earliest publication to use double factorial notation)〔 states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals arising in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, and they have many applications in enumerative combinatorics.〔〔 The term odd factorial is sometimes used for the double factorial of an odd number.〔E.g., in and .〕 ==Relation to the factorial== Because the double factorial only involves about half the factors of the ordinary factorial, its value is not substantially larger than the square root of the factorial ''n''!, and it is much smaller than the iterated factorial (''n''!)!. For an even positive integer ''n'' = 2''k'', ''k'' ≥ 0, the double factorial may be expressed as :$n!!\; =\; 2^k\; k!.$ For odd ''n'' = 2''k'' − 1, ''k'' ≥ 1, it has the expressions :$n!!\; =\; \backslash frac\; =\; \backslash frac.$ In this expression, the first denominator equals (2''k'')!! and cancels the unwanted even factors from the numerator. For an odd positive integer ''n'' = 2''k'' − 1, ''k'' ≥ 1, the double factorial may be expressed in terms of ''k''-permutations of ''2k'' as〔〔.〕 :$(2k-1)!!\; =\; \backslash frac\; =\; \backslash frac\; .$ 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「Double factorial」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.