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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 Pentagonal number theorem ： ウィキペディア英語版 Pentagonal number theorem In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that :$\backslash prod\_^\backslash left(1-x^\backslash right)=\backslash sum\_^\backslash left(-1\backslash right)^x^\backslash left(3k-1\backslash right)\}=1+\backslash sum\_^\backslash infty(-1)^k\; \backslash left(1+x^k\backslash right)x^=\backslash sum\_^\backslash infty(-1)^k\; \backslash left(1-x^\backslash right)x^\; .$ In other words, :$(1-x)(1-x^2)(1-x^3)\; \backslash cdots\; =\; 1\; -\; x\; -\; x^2\; +\; x^5\; +\; x^7\; -\; x^\; -\; x^\; +\; x^\; +\; x^\; -\; \backslash cdots.$ The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula for ''k'' = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers. This holds as an identity of convergent power series for , and also as an identity of formal power series. A striking feature of this formula is the amount of cancellation in the expansion of the product. == Relation with partitions == The identity implies a marvelous recurrence for calculating $p(n)$, the number of partitions of ''n'': :$p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\backslash cdots$ or more formally, :$p(n)=\backslash sum\_k\; (-1)^p(n-g\_k)$ where the summation is over all nonzero integers ''k'' (positive and negative) and $g\_k$ is the ''k''th pentagonal number. 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「Pentagonal number theorem」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.