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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 Leibniz harmonic triangle ： ウィキペディア英語版 Leibniz harmonic triangle The Leibniz harmonic triangle is a triangular arrangement of fractions in which the outermost diagonals consist of the reciprocals of the row numbers and each inner cell is the absolute value of the cell above minus the cell to the left. To put it algebraically, (where is the number of the row, starting from 1, and is the column number, never more than ''r'') and The first eight rows are: $\backslash begin$ & & & & & & & & & 1 & & & & & & & &\\ & & & & & & & & \frac & & \frac & & & & & & &\\ & & & & & & & \frac & & \frac & & \frac & & & & & &\\ & & & & & & \frac & & \frac & & \frac & & \frac & & & & &\\ & & & & & \frac & & \frac & & \frac & & \frac & & \frac & & & &\\ & & & & \frac & & \frac & & \frac & & \frac & & \frac & & \frac & & &\\ & & & \frac & & \frac & & \frac & & \frac & & \frac & & \frac & & \frac & &\\ & & \frac & & \frac & & \frac & & \frac & & \frac & & \frac & & \frac & & \frac &\\ & & & & &\vdots & & & & \vdots & & & & \vdots& & & & \\ \end The denominators are listed in , while the numerators are all 1s. Whereas each entry in Pascal's triangle is the sum of the two entries in the above row, each entry in the Leibniz triangle is the sum of the two entries in the row ''below'' it. For example, in the 5th row, the entry (1/30) is the sum of the two (1/60)s in the 6th row. Just as Pascal's triangle can be computed by using binomial coefficients, so can Leibniz's: $L(r,\; c)\; =\; \backslash frac$. For example, for the 3rd row, we have 3 + 6 + 3 = 12 = 3 × 22. It is worth noting that $L(r,\; c)\; =\; \backslash int\_0^1\; \backslash !\; x\; ^\; (1\; -\; x)^\; \backslash ,dx\; \backslash ,.$ ==References== 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「Leibniz harmonic triangle」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.