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A266770
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Molien series for invariants of finite Coxeter group D_7.
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10
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1, 0, 1, 0, 2, 0, 3, 1, 5, 1, 7, 2, 11, 3, 15, 5, 21, 7, 28, 11, 38, 15, 49, 21, 65, 28, 82, 38, 105, 49, 131, 65, 164, 82, 201, 105, 248, 131, 300, 164, 364, 201, 436, 248, 522, 300, 618, 364, 733, 436, 860, 522, 1009, 618, 1175, 733, 1367, 860, 1579, 1009, 1824, 1175, 2093, 1367, 2400, 1579, 2738, 1824, 3120, 2093, 3539, 2400, 4011
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OFFSET
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0,5
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COMMENTS
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The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
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REFERENCES
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J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,0,1,0,-1,-1,-1,0,0,-2,0,0,1,1,0,1,2,1,0,1,-1, 0,-1,-2,-1,0,-1,-1,0,0,2,0,0,1,1,1,0,-1,0,0,-1,0,-1,0,1).
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FORMULA
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G.f.: 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^8)*(1-x^10)*(1-x^12)).
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MAPLE
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seq(coeff(series(1/((1-x^7)*mul(1-x^(2*j), j=1..6)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Jan 31 2020
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MATHEMATICA
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CoefficientList[Series[1/((1-x^7)*Product[1-x^(2*j), {j, 6}]), {x, 0, 80}], x] (* G. C. Greubel, Jan 31 2020 *)
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PROG
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(PARI) Vec(1/((1-x^7)*prod(j=1, 6, 1-x^(2*j))) +O('x^80)) \\ G. C. Greubel, Jan 31 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^7)*(&*[1-x^(2*j): j in [1..6]])) )); // G. C. Greubel, Jan 31 2020
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/((1-x^7)*product(1-x^(2*j) for j in (1..6))) ).list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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