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%I #36 Feb 12 2024 13:24:47
%S 1,5,14,30,55,91,140,204,285,385,506,651,823,1024,1256,1521,1821,2158,
%T 2534,2952,3415,3925,4485,5098,5766,6491,7275,8120,9028,10002,11046,
%U 12162,13351,14616,15960,17385,18893,20486,22167,23939,25805,27768,29829,31989,34251,36618,39092,41675,44370,47180,50106,53150,56315,59602,63012
%N The growth series for the affine Coxeter (or Weyl) group [3,3,5] (or H_4).
%D 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).
%D H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial; also Table 3.1 page 59.
%H Vincenzo Librandi, <a href="/A266783/b266783.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_60">Index entries for linear recurrences with constant coefficients</a>, 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).
%F 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).
%F 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
%p 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
%t 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 *)
%o (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
%o (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
%o (Sage)
%o def A266783_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o 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()
%o A266783_list(60) # _G. C. Greubel_, Feb 04 2020
%Y For the growth series for the finite group see A162497.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 11 2016
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