A008582 Molien series for Weyl group E_8.

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, 111, 126, 146, 163, 187, 211, 238, 266, 302, 335, 378, 421, 469, 520, 582, 640, 712, 786, 868, 954, 1055, 1153, 1270, 1391, 1523, 1662, 1822, 1979, 2162, 2352, 2558, 2774, 3018, 3262

Offset

0,5

References

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

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

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 252

Index entries for Molien series

Formula

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)).

a(n) ~ 1/13716864000*n^7 (for the sequence without interleaved zeros). - Ralf Stephan, Apr 29 2014

Maple

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

Mathematica

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 *)
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 *)

Prog

(Magma) MolienSeries(CoxeterGroup("E8")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(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
(Sage)
def A008582_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( product(1/((1-x^(3*j+6))*(1-x^(3*j+1))) for j in (0..3)) ).list()
A008582_list(60) # G. C. Greubel, Feb 02 2020

Crossrefs

Sequence in context: A319069 A029013 A114096 * A069911 A185225 A027196

Adjacent sequences: A008579 A008580 A008581 * A008583 A008584 A008585

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved