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%I #20 Feb 12 2024 13:22:17
%S 1,4,9,16,25,37,52,69,88,110,136,165,196,229,265,304,345,388,434,484,
%T 537,592,649,709,772,837,904,974,1048,1125,1204,1285,1369,1456,1545,
%U 1636,1730,1828,1929,2032,2137,2245,2356,2469,2584,2702,2824,2949,3076,3205,3337,3472,3609,3748,3890,4036,4185,4336,4489,4645
%N The growth series for the affine Coxeter (or Weyl) group [3,5] (or H_3).
%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="/A266782/b266782.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 0, 3, -3, 0, 3, -3, 1).
%F G.f.: (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2).
%p m:= 60; S:=series((1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 04 2020
%t CoefficientList[Series[(1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2), {x, 0, 60}], x] (* _Vincenzo Librandi_, Sep 07 2016 *)
%o (PARI) Vec( (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2) +O('x^60) ) \\ _G. C. Greubel_, Feb 04 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2) )); // _G. C. Greubel_, Feb 04 2020
%o (Sage)
%o def A266782_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2) ).list()
%o A266782_list(60) # _G. C. Greubel_, Feb 04 2020
%Y For the growth series for the finite group see A162495.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 11 2016
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