|
%I #41 Sep 08 2022 08:44:36
%S 1,1,1,2,3,4,6,8,10,14,18,22,29,36,44,55,67,80,98,117,138,165,194,226,
%T 266,309,356,413,475,542,622,708,802,911,1029,1157,1304,1462,1633,
%U 1827,2036,2261,2514,2785
%N Molien series for Weyl group E_7.
%C The relevant generating function 1/((1-z^2)*(1-z^6)*(1-z^8)*(1-z^10)*(1-z^12)*(1-z^14)*(1-z^18)) is reduced with z^2=x below to indicate that the intermediate zeros are not stored in this sequence.
%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. 36).
%H T. D. Noe, <a href="/A008583/b008583.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=250">Encyclopedia of Combinatorial Structures 250</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_35">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, 0, 0, 0, -1, -1, 0, -1, 0, 1, 0, 2, 0, 1, 0, 0, -1, 0, -2, 0, -1, 0, 1, 0, 1, 1, 0, 0, 0, -1, 0, -1, 1).
%F G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^9)).
%p A008583_list := proc(n) local G,j;
%p G:= series(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^9)),x,n+1);
%p [seq(coeff(G,x,j),j=0..n)];
%p end proc; # _Robert Israel_, Mar 26 2012
%t CoefficientList[Series[1/((1-x)(1-x^3)(1-x^4)(1-x^5)(1-x^6)(1-x^7)(1-x^9)),{x,0,50}],x] (* _Harvey P. Dale_, Mar 04 2013 *)
%o (Magma) MolienSeries(CoxeterGroup("E7")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o (PARI) A008583_list(n)=Vec(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^9))+O(x^n)) /* returns n terms [a(0),...,a(n-1)] */ \\ _M. F. Hasler_, Mar 26 2012
%o (Sage)
%o def A008583_list(n) :
%o R.<t> = PowerSeriesRing(ZZ)
%o G = 1/((1-t)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^9) + O(t^n))
%o return G.padded_list() # _Peter Luschny_, Mar 27 2012
%Y Cf. A005795.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
|
