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I was wondering if there is a closed formula for sum of cubed combinations. More precisely, I'd like to compute $$\sum_{k=1}^n \left ( \begin{array}{c}n\\k\end{array}\right )^3$$

Obviously, without the "$^3$", the sum is $2^n-1$....

Any idea ?

thanks

dcoeurjo
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  • Agree.. thanks a lot, I've missed this post – dcoeurjo May 14 '13 at 15:09
  • I'm glad to help. – MJD May 14 '13 at 15:10
  • Do I have something to do to "close" this question or the "duplicate" is enough ? (newbie on stackexchange) – dcoeurjo May 14 '13 at 15:11
  • @dcoeurjo: It needs four or five votes to close as a duplicate. The remaining should be here soon. Thanks. – Ross Millikan May 14 '13 at 15:17
  • http://math.stackexchange.com/questions/134747/sum-of-cubes-of-binomial-coefficients – lab bhattacharjee May 14 '13 at 17:47
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    @dcoeurjo: Is this really a duplicate? The other question asks only for an asymptotic estimate, not a closed form. – robjohn May 14 '13 at 19:19
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    @labbhattacharjee: that question asks for an asymptotic estimate, whereas this question seeks a closed form (which usually means exact). – robjohn May 14 '13 at 19:21
  • @robjohn Thanks. Why do you think it should be reopened? I did observe that the other question asked only for an asymptotic estimate, but I voted to close because the answers addressed dcoeurjo's request for a closed form. – MJD May 14 '13 at 19:24
  • @robjohn After considering further, I now agree with you. There is room on the site for a more extensive discussion of the nonexistence of a closed form for this sum. I will vote to reopen. – MJD May 14 '13 at 19:31
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    I originally voted to close this, but the question is sufficiently different that I have nominated it for reopening. The other, similar question asks for an approximation formula. The answers there mention that no closed-form exists, and cite a paper of Farmer and Leth, which does not give the proof. I think further discussion of this question is warranted, and our site could benefit from a more detailed discussion of the nonexistence of a closed form for this sum. – MJD May 14 '13 at 19:33
  • @robjohn As my comment above indicates, I agree for essentially the same reasons. (Also see here.) I have nominated the question for reopening. – MJD May 14 '13 at 19:37

1 Answers1

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These are almost the same as the Franel numbers - just replace the lower limit on $k$ with 0 instead of 1 ($n\choose 0$ is always 1, so this is just a simple shift of the resulting sequence). While there is a closed form expression for the Franel numbers, it requires a generalised hypergeometric function, so it isn't enormously satisfying:

$$ \sum_{k=0}^n {n \choose k}^3 = {}_3F_2 (-n,-n,-n;1,1;-1) $$

TCL
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Martin O'Leary
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