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Search: id:A077415
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| A077415 |
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Number of independent components of a certain 3-tensor in n-space. |
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+0
3
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| 0, 5, 16, 35, 64, 105, 160, 231, 320, 429, 560, 715, 896, 1105, 1344, 1615, 1920, 2261, 2640, 3059, 3520, 4025, 4576, 5175, 5824, 6525, 7280, 8091, 8960, 9889, 10880, 11935, 13056, 14245, 15504, 16835, 18240, 19721, 21280, 22919, 24640, 26445
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c)=t(b,a,c) and sum(t(a,a,c),a=1..n)=0 for all c, and t(a,b,c)+t(b,c,a)+t(c,a,b)=0, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices and satisfies the cyclic identity.)
Number of standard tableaux of shape (n-1,2,1) (n>=3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004
a(n) = A084990(n - 1) - 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2007
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FORMULA
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a(n)= n*(n+2)*(n-2)/3 = A077414(n)-binomial(n+2, 3). binomial(n+2, 3)=A000292(n-1).
G.f.: x^3*(5-4*x+x^2)/(1-x)^4.
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MAPLE
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seq (((n^3)-4*n)/3, n=2..35); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 20 2007
a:=n->sum(sum(sum(5, j=0..n), k=2..n), m=4..n)/15: seq(a(n), n=3..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 25 2007
seq(sum(n^2-4, k=1..n)/3, n=2..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 28 2008
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CROSSREFS
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Sequence in context: A131425 A096941 A098404 this_sequence A108966 A072333 A055232
Adjacent sequences: A077412 A077413 A077414 this_sequence A077416 A077417 A077418
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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EXTENSIONS
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More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2007
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