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who could kindly give me the formula for $$(x_1+x_2 + \cdots+ x_n) ^3,$$ in the form like the case $$(x_1+x_2 + \cdots+ x_n) ^2 = \sum^n_{i=1} x_i^2 + 2\sum_{1\leq i<j\leq n} x_ix_j.$$

Thanks

Sean
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1 Answers1

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$$\sum_i x_i^3+3\sum_{i\neq j}x_i^2 x_j+6\sum_{\substack{\{i,j,k\}\\i\neq j\neq k}}x_ix_jx_k$$

Bernard
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  • thank you, that's exactly what I want – Sean Aug 12 '15 at 18:18
  • Though it's long time ago, but shouldn't $i \ne j $ and $i \ne j \ne k$ be "$ i < j$" and "$i < j < k", respectively? – Sean Nov 26 '15 at 02:24
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    Again a long time but I think it should be $i \neq j$ and $i<j<k$? – user103828 Jan 12 '20 at 20:56
  • Yes, you're right. It was implicit in my mind. However, I prefer the way I've modified it, because it has the same form as the previous sum. Anyway, thank you for pointing it! – Bernard Jan 12 '20 at 21:12