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A006584
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If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.
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5
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0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Graded dimension of L''/[L',L''] for the free Lie algebra on 2 generators. Let L be a free Lie algebra with 2 generators graded by the total degree. Set L'=[L,L] and L''=[L',L']. Then a(n) is equal to the dimension of the homogeneous subspace of degree n+2 in the quotient L''/[L',L'']. - Sergei Duzhin (duzhin(AT)pdmi.ras.ru), Mar 15 2004
a(n+3)= A003451(n)+A027656(n) - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008
Also the 2nd Witt transform of A000027. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
Also the number of 3-element subsets of {1,...,n+1} whose elements sum up to an odd integer, i.e. the third column of A159916: e.g. a(3)=2 corresponds to the two subsets {1,2,4} and {2,3,4} of {1,...,4}. [From M. F. Hasler (www.univ-ag.fr/~mhasler), May 01 2009]
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REFERENCES
| W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.
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LINKS
| Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO]. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
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FORMULA
| G.f.: 2x^3/((1-x)^4*(1+x)^2). a(n)=2*A006918(n-2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
Linear recurrence: a(n)=2a(n-1)+a(n-2)-4a(n-3)+a(n-4)+2a(n-5)-a(n-6) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]
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PROG
| (PARI) A006584(n)=n*(n^2-if(n%2, 1, 4))\12 [From M. F. Hasler (www.univ-ag.fr/~mhasler), May 01 2009]
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CROSSREFS
| Cf. A003451, A027656, A034877 .
Sequence in context: A123689 A137928 A144834 * A032246 A141138 A077627
Adjacent sequences: A006581 A006582 A006583 * A006585 A006586 A006587
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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