A008670 Molien series for Weyl group F_4.

1, 1, 1, 2, 3, 3, 5, 6, 7, 9, 11, 12, 16, 18, 20, 24, 28, 30, 36, 40, 44, 50, 56, 60, 69, 75, 81, 90, 99, 105, 117, 126, 135, 147, 159, 168, 184, 196, 208, 224, 240, 252, 272, 288, 304, 324, 344, 360, 385, 405, 425, 450, 475, 495, 525, 550, 575, 605, 635, 660, 696, 726, 756

Offset

0,4

Comments

Number of partitions of n into parts 1, 3, 4 and 6. - Ilya Gutkovskiy, May 24 2017

References

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

L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 28).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 236

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1).

Formula

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)). [Corrected by Ralf Stephan, Apr 29 2014]

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-11) + a(n-13) - a(n-14), with a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(6)=5, a(7)=6, a(8)=7, a(9)=9, a(10)=11, a(11)=12, a(12)=16, a(13)=18. - Harvey P. Dale, Feb 07 2012

a(n) ~ (1/432)*n^3. - Ralf Stephan, Apr 29 2014

a(n) = (120*floor(n/6)^3 + 60*(m+7)*floor(n/6)^2 + 2*(m^5-15*m^4+75*m^3-135*m^2+134*m+240)*floor(n/6) + 3*(m^5-15*m^4+75*m^3-135*m^2+84*m+70) + (m^5-15*m^4+75*m^3-135*m^2+44*m+30)*(-1)^floor(n/6))/240 where m = (n mod 6). - Luce ETIENNE, Aug 14 2018

Maple

a:= proc(n) local m, r; m := iquo (n, 12, 'r'); r:= r+1; ([4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26][r]+ (6+r+4*m)*m)*m+ [1$3, 2, 3$2, 5, 6, 7, 9, 11, 12][r] end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 06 2008

Mathematica

Take[CoefficientList[Series[1/((1-x^2)(1-x^6)(1-x^8)(1-x^12)), {x, 0, 130}], x], {1, -1, 2}] (* or *) LinearRecurrence[ {1, 0, 1, 0, -1, 1, -2, 1, -1, 0, 1, 0, 1, -1}, {1, 1, 1, 2, 3, 3, 5, 6, 7, 9, 11, 12, 16, 18}, 70] (* Harvey P. Dale, Feb 07 2012 *)

Prog

(Magma) MolienSeries(CoxeterGroup("F4")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 08 2019
(PARI) my(x='x+O('x^70)); Vec(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))) \\ G. C. Greubel, Sep 08 2019
(Sage)
def A008670_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))).list()
A008670_list(70) # G. C. Greubel, Sep 08 2019

Crossrefs

Cf. A002411, A006002, A010875, A011934, A027480, A055232, A182260.

Sequence in context: A236294 A251419 A036410 * A338887 A338886 A218950

Adjacent sequences: A008667 A008668 A008669 * A008671 A008672 A008673

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved