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%I #20 Feb 12 2024 13:26:13
%S 1,5,14,30,55,92,144,214,305,419,559,729,933,1173,1451,1771,2137,2552,
%T 3018,3537,4111,4744,5441,6205,7037,7940,8919,9977,11116,12338,13646,
%U 15043,16533,18120,19805,21590,23480,25480,27592,29817,32158,34618,37200,39908,42745,45713,48815,52056,55439,58965,62637,66459,70434,74564,78852,83301
%N The growth series for the affine Weyl group F_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 G. C. Greubel, <a href="/A266784/b266784.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1).
%F G.f.: (1+t)*(1+t+t^2+t^3+t^4+t^5)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(1+t+t^2+t^3 +t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11)/((1-t)*(1-t^5)*(1-t^7)*(1-t^11)).
%F G.f.: (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)). - _G. C. Greubel_, Feb 04 2020
%p m:=30; S:=series((1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 04 2020
%t CoefficientList[Series[(1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)), {x,0,60}], x] (* _G. C. Greubel_, Feb 04 2020 *)
%o (PARI) Vec( (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)) +O('x^60) ) \\ _G. C. Greubel_, Feb 04 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)) )); // _G. C. Greubel_, Feb 04 2020
%o (Sage)
%o def A077952_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)) ).list()
%o A077952_list(60) # _G. C. Greubel_, Feb 04 2020
%Y For the growth series for the finite group see A162496.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 11 2016
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