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A122391
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Dimension of 2-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 2 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).
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5
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1, 1, 1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Except for first couple of terms, series agrees with A003945
a(n) written in base 2: a(0) = 1, a(1) = 1, a(2) = 1, a(n) for n >= 3: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-3) times 0 (see A003953(n-2). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 17 2009]
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REFERENCES
| N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, to appear Canad. J. Math., arXiv:math.CO/0502082
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
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
| G.f.: (1-q)*(1-q^2)/(1-2*q) 2^n - 2^(n-1) - 2^(n-2) + 2^(n-3) (for n>2) a(0) = 1, a(1) = 1, a(2) = 1, a(n) = 3*2^(n-3) for n>2
a(n) = 3*2^(n-3) = 2^(n-3) + 2^(n-2) for n >= 3. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 17 2009]
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EXAMPLE
| a(1) = 1 because x1 - x2 is killed by d_x1 + d_x2
a(2) = 1 because x1 x2 - x2 x1 is killed by d_x1+d_x2, d_x1^2 + d_x2^2
a(3) = 3 because x1 x1 x2 - 2 x1 x2 x1 + x2 x1 x1, x1 x2 x2 - 2 x2 x1 x2 + x2 x2 x1, x1 x1 x2 - x1 x2 x1 - x2 x1 x2 + x2 x2 x1 are all killed by d_x1 + d_x2, d_x1^2 + d_x2^2, d_x1 d_x2, d_x1^3 + d_x2^3 and d_x1^2 d_x2 + d_x1 d_x2^2
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MAPLE
| coeffs(convert(series((1-q)*(1-q^2)/(1-2*q), q, 20), `+`)-O(q^20), q);
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CROSSREFS
| Cf. A003945, A034008, A122367, A122392, A122393, A122394.
Sequence in context: A169064 A169112 A169160 * A169208 A169256 A169304
Adjacent sequences: A122388 A122389 A122390 * A122392 A122393 A122394
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KEYWORD
| nonn
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AUTHOR
| Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 31 2006
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