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A266783
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The growth series for the affine Coxeter (or Weyl) group [3,3,5] (or H_4).
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1
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1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 651, 823, 1024, 1256, 1521, 1821, 2158, 2534, 2952, 3415, 3925, 4485, 5098, 5766, 6491, 7275, 8120, 9028, 10002, 11046, 12162, 13351, 14616, 15960, 17385, 18893, 20486, 22167, 23939, 25805, 27768, 29829, 31989, 34251, 36618, 39092, 41675, 44370, 47180, 50106, 53150, 56315, 59602, 63012
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OFFSET
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0,2
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REFERENCES
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N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial; also Table 3.1 page 59.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
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FORMULA
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G.f. = t1/t2 where t1 is (1 + t)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10 + t^11)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10 + t^11 + t^12 + t^13 + t^14 + t^15 + t^16 + t^17 + t^18 + t^19)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10 + t^11 + t^12 + t^13 + t^14 + t^15 + t^16 + t^17 + t^18 + t^19 + t^20 + t^21 + t^22 + t^23 + t^24 + t^25 + t^26 + t^27 + t^28 + t^29) and t2 = (1 - t)*(1 - t^11)*(1 - t^19)*(1 - t^29).
G.f.: (1 - x^2)*(1 - x^12)*(1 - x^20)*(1 - x^30)/((1 - x)^5*(1 - x^11)*(1 - x^19)*(1 - x^29)). - G. C. Greubel, Feb 04 2020
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MAPLE
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m:=60; S:=series((1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 04 2020
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MATHEMATICA
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CoefficientList[Series[(1 + t)^4 * (1 + t^2) * (1 + t^2 + t^4) * (1 + t^4 + t^8) * (1 + t^2 + t^4 + t^6 + t^8) * (1 + t^6 + t^10 + t^12 + t^16 + t^18 + t^22 + t^24 + t^28 + t^34)/((1 - t) * (1 - t^11) * (1 - t^19) * (1 - t^29)), {t, 0, 60}], t] (* Wesley Ivan Hurt, Apr 12 2017; modified by G. C. Greubel, Feb 04 2020 *)
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PROG
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(PARI) Vec( (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)) +O('x^60) ) \\ G. C. Greubel, Feb 04 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)) )); // G. C. Greubel, Feb 04 2020
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)) ).list()
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CROSSREFS
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For the growth series for the finite group see A162497.
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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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