login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008670 Molien series for Weyl group F_4. 2
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 (list; graph; refs; listen; history; text; internal format)
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 * A218950 A193748 A322526

Adjacent sequences:  A008667 A008668 A008669 * A008671 A008672 A008673

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 31 18:30 EDT 2020. Contains 333151 sequences. (Running on oeis4.)