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Search: id:A118264
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| A118264 |
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Coefficient of q^n in (1-q)^3/(1-3q); dimensions of the enveloping algebra of the derivied free Lie algebra on 3 letters. |
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4
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| 1, 0, 3, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, http://arXiv.org/abs/math.CO/0502082, to appear Canad. Math J.
C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.
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FORMULA
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g.f. (1-q)^3/(1-3q); sum( (-1)^k*C(2,k) 3^k; k=0..n)
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EXAMPLE
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the enveloping algebra of the derivied free Lie algebra is characterized as the interesection of the kernels of all partial derivative operators in the space of non-commutative polynomials, a(0) = 1 since all constants are killed by derivatives, a(1) = 0 since no polys of degree 1 are killed, a(2) = 3 since all Lie brackets [x1,x2], [x1,x3], [x2, x3] are killed by all derivative operators
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MAPLE
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f:=n->coeftayl((1-q)^3/(1-3*q), q=0, n):seq(f(i), i=0..15);
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CROSSREFS
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Cf. A080923, A027376, A118265, A118266.
Sequence in context: A052855 A133787 A080923 this_sequence A006365 A046919 A046342
Adjacent sequences: A118261 A118262 A118263 this_sequence A118265 A118266 A118267
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KEYWORD
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nonn
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AUTHOR
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Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Apr 20 2006
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