I want to know how to caculate $\binom{2n}{0}^3-\binom{2n}{1}^3+\cdots+(-1)^k\binom{2n}{k}^3 \cdots+\binom{2n}{2n}^3$? The sum equals $ (-1)^{n}\binom{3n}{2n}\binom{2n}{n} $, but I donot know how to get this.


  • $\begingroup$ I was about to help with your latest question on irreducibles, but could not because it was deleted. This is not nice to all those who spent valuable time thinking about it. $\endgroup$ – Bill Dubuque Apr 9 '12 at 17:03
  • $\begingroup$ Why not reopen it, and rephrase the question as follows: can an irreducible element become reducible in a localization? This uses the standard definition of irreducible, so avoids the distracting comments about units and nonstandard definitions. This is a good question, so deserves to answered. $\endgroup$ – Bill Dubuque Apr 9 '12 at 17:14
  • $\begingroup$ @BillDubuque, thank you. As someone said that this is an easy exercise, so I try some example. e.g. $A=\mathbb{Z}[\sqrt{-5}]$, then $2$ is irreducible in $A$, but $2$ is reducible in $A_3$. $\endgroup$ – wxu Apr 9 '12 at 17:18
  • $\begingroup$ @BillDubuque, I donot know how to reopen it. I have voted to delete it. Then it disappeared.. $\endgroup$ – wxu Apr 9 '12 at 17:20
  • $\begingroup$ In such cases it is better to answer your own question. That way you can get feedback on your answer, and may learn other answers too. $\endgroup$ – Bill Dubuque Apr 9 '12 at 17:35

Look for Dixon's Identity. It is among other places discussed in the following Wikipedia article


See Recurrences for alternating sums of powers of binomial coefficients, which cites Dixon's Summation of a certain series.


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