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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Cf. A014977.

Programs

Magma

MolienSeries(CoxeterGroup("E6")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

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 *)

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. = 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

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