A008584 Molien series for Weyl group E_6.

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, 43, 41, 54, 49, 66, 61, 78, 75, 95, 89, 113, 107, 132, 129, 157, 150, 184, 178, 212, 209, 248, 241, 287, 280, 327

Offset

0,7

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.

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.

L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 35).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 248

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, 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).

Formula

G.f.: 1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)).

a(n) ~ 1/6220800*n^5 + 1/414720*n^4. - Ralf Stephan, Apr 29 2014

Maple

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

Mathematica

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 *)

Prog

(Magma) MolienSeries(CoxeterGroup("E6")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(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
(Sage)
def A008584_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)) ).list()
A008584_list(60) # G. C. Greubel, Jan 31 2020

Crossrefs

Cf. A014977.

Sequence in context: A115584 A058742 A029140 * A352833 A034390 A368671

Adjacent sequences: A008581 A008582 A008583 * A008585 A008586 A008587

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved