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Does anybody know where I can find a general form, in terms of n, of the sum $\left(\sum\limits_{i=1}^n a_i\right)^n$. What I mean is, there appears to be some sort of pattern, if you look at $\left(\sum\limits_{i=1}^2 a_i\right)^2$ you get $a_1^2+2a_1a_2+a_2^2$ and if you let $n=3$ then you get $a_1^3+a_2^3+a_3^3+3a_1^2(a_2+a_3)+3a_2^2(a_1+a_3)+3a_3^2(a_2+a_3)+6a_1a_2a_3$. I'm working with something right now where I'd like to subtract out the cubed terms and have a general formula left over for the remaining terms which I can write simply in terms of $n$. Sorry about the lack of clarity, I hope this helps.

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Your question is a bit vague. I have added an answer which I think might be what you are looking for, but I suggest you make your question more specific and perhaps add a few examples... –  Aryabhata Dec 3 '10 at 6:55
    
The actual ultimate goal here is to show that $\vert a\vert_n\leq\vert a\vert_1\leq\sqrt[n]{n}\vert a\vert_n$ where $a$ is in $\mathbb{R}^n$ and $\vert a\vert_k$ is just basically the $\ell^k$ norm applied to this finite sequence. –  Jon Beardsley Dec 3 '10 at 7:09
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up vote 6 down vote accepted

It isn't really clear what you are asking.

Perhaps this will help: http://en.wikipedia.org/wiki/Multinomial_theorem

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I think this multinomial formula would work. Can't believe I forgot this from undergrad combinatorics, oh well. –  Jon Beardsley Dec 3 '10 at 7:04
    
That's a pretty good answer, but I'm going to have to reformulate my entire problem anyway, this is far too tedious dealing with all of these summations. –  Jon Beardsley Dec 3 '10 at 7:23
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