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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 Stirling transform ： ウィキペディア英語版 Stirling transform In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by :$b\_n=\backslash sum\_^n\; \backslash left\backslash \; \backslash right\backslash \}\; a\_k,$ where $\backslash left\backslash \; \backslash right\backslash \}$ is the Stirling number of the second kind, also denoted ''S''(''n'',''k'') (with a capital ''S''), which is the number of partitions of a set of size ''n'' into ''k'' parts. The inverse transform is :$a\_n=\backslash sum\_^n\; s(n,k)\; b\_k,$ where ''s''(''n'',''k'') (with a lower-case ''s'') is a Stirling number of the first kind. Berstein and Sloane (cited below) state "If ''a''''n'' is the number of objects in some class with points labeled 1, 2, ..., ''n'' (with all labels distinct, i.e. ordinary labeled structures), then ''b''''n'' is the number of objects with points labeled 1, 2, ..., ''n'' (with repetitions allowed)." If :$f(x)=\backslash sum\_^\backslash infty\; x^n$ is a formal power series (note that the lower bound of summation is 1, not 0), and :$g(x)=\backslash sum\_^\backslash infty\; x^n$ with ''a''''n'' and ''b''''n'' as above, then :$g(x)=f(e^x-1).\backslash ,$ ==See also== * Binomial transform * List of factorial and binomial topics 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「Stirling transform」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.