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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 Trinomial triangle ： ウィキペディア英語版 Trinomial triangle The trinomial triangle is a variation of Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the ''three'' (rather than the ''two'' in Pascal's triangle) entries above it: $\backslash begin$ & & & & 1\\ & & & 1& 1&1\\ & & 1& 2& 3&2&1\\ &1& 3& 6& 7&6&3&1\\ 1&4&10&16&19&16&10&4&1\end The $k$-th entry of the $n$-th row is denoted by : $\_2$. Rows are counted starting from 0. The entries of the $n$-th row are indexed starting with $-n$ from the left, and the middle entry has index 0. The symmetry of the entries of a row about the middle entry is expressed by the relationship : $\_2=\_2$ == Properties == The $n$-th row corresponds to the coefficients in the polynomial expansion of the expansion of the trinomial $(1\; +\; x\; +\; x^2)$ raised to the $n$-th power: :$\backslash left(1+x+x^2\backslash right)^n=\; \backslash sum\; \_^\_2\; x^=\backslash sum\; \_^\_2\; x^$ or, symmetrically, :$\backslash left(1+x+1/x\backslash right)^n=\backslash sum\_^\_2\; x^k$, hence the alternative name trinomial coefficients because of their relationship to the multinomial coefficients: : $\_2=\backslash sum\_$ Furthermore, the diagonals have interesting properties, such as their relationship to the triangular numbers. The sum of the elements of $n$-th row is $3^n$. 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「Trinomial 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.