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%I #32 Aug 14 2022 16:47:51
%S 1,0,1,0,1,1,2,1,3,2,4,3,6,4,8,6,10,9,14,11,18,15,22,20,29,25,36,32,
%T 43,41,54,49,66,61,78,75,95,89,113,107,132,129,157,150,184,178,212,
%U 209,248,241,287,280,327
%N Molien series for Weyl group E_6.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
%D H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und Ihrer Grenzgebiete, New Series, no.14. Springer Verlag, 1957, Table 10.
%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 35).
%H T. D. Noe, <a href="/A008584/b008584.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=248">Encyclopedia of Combinatorial Structures 248</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_42">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,0,1,1,-1,0,1,-1,-2,1,0,-3,0,2,-1,-1,3,1,-2,1,3,-1,-1,2,0,-3,0,1,-2,-1,1,0,-1,1,1,0,0,1,0,-1).
%F G.f.: 1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)).
%F a(n) ~ 1/6220800*n^5 + 1/414720*n^4. - _Ralf Stephan_, Apr 29 2014
%p seq(coeff(series(1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)), x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Jan 31 2020
%t CoefficientList[Series[1/((1-x^2)(1-x^5)(1-x^6)(1-x^8)(1-x^9)(1-x^12)),{x,0,55}],x] (* _Harvey P. Dale_, Aug 10 2011 *)
%o (Magma) MolienSeries(CoxeterGroup("E6")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o (PARI) my(x='x+O('x^60)); Vec(1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12))) \\ _G. C. Greubel_, Jan 31 2020
%o (Sage)
%o def A008584_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)) ).list()
%o A008584_list(60) # _G. C. Greubel_, Jan 31 2020
%Y Cf. A014977.
%K nonn,easy,nice
%O 0,7
%A _N. J. A. Sloane_
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