A008582 - OEIS

%I #32 Sep 08 2022 08:44:36

%S 1,1,1,1,2,2,3,4,5,6,8,9,12,14,17,20,25,28,34,40,47,54,64,72,85,97,

%T 111,126,146,163,187,211,238,266,302,335,378,421,469,520,582,640,712,

%U 786,868,954,1055,1153,1270,1391,1523,1662,1822,1979,2162,2352,2558,2774,3018,3262

%N Molien series for Weyl group E_8.

%D Coxeter and Moser, Generators and Relations for Discrete Groups, Table 10.

%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 37).

%H T. D. Noe, <a href="/A008582/b008582.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=252">Encyclopedia of Combinatorial Structures 252</a>

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%F G.f.: 1/((1-x^2)*(1-x^8)*(1-x^12)*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^24)*(1-x^30)).

%F a(n) ~ 1/13716864000*n^7 (for the sequence without interleaved zeros). - _Ralf Stephan_, Apr 29 2014

%p seq(coeff(series( mul(1/((1-x^(3*j+6))*(1-x^(3*j+1))), j=0..3), x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Feb 02 2020

%t Select[CoefficientList[Series[1/((1-x^2)(1-x^8)(1-x^12)(1-x^14)(1-x^18) (1-x^20)(1-x^24)(1-x^30)),{x,0,180}],x],#!=0&] (* _Harvey P. Dale_, Jun 09 2011 *)

%t CoefficientList[Series[Product[1/((1-x^(3*j+6))*(1-x^(3*j+1))), {j,0,3}], {x, 0, 60}], x] (* _G. C. Greubel_, Feb 02 2020 *)

%o (Magma) MolienSeries(CoxeterGroup("E8")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

%o (PARI) Vec( prod(j=0,3, 1/((1-x^(3*j+6))*(1-x^(3*j+1)))) +O('x^60) ) \\ _G. C. Greubel_, Feb 02 2020

%o (Sage)

%o def A008582_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( product(1/((1-x^(3*j+6))*(1-x^(3*j+1))) for j in (0..3)) ).list()

%o A008582_list(60) # _G. C. Greubel_, Feb 02 2020

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_