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A008670 Molien series for Weyl group F_4. 2

%I

%S 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,

%T 81,90,99,105,117,126,135,147,159,168,184,196,208,224,240,252,272,288,

%U 304,324,344,360,385,405,425,450,475,495,525,550,575,605,635,660,696,726,756

%N Molien series for Weyl group F_4.

%C Number of partitions of n into parts 1, 3, 4 and 6. - _Ilya Gutkovskiy_, May 24 2017

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

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

%H Alois P. Heinz, <a href="/A008670/b008670.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=236">Encyclopedia of Combinatorial Structures 236</a>

%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1).

%F G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)). [Corrected by _Ralf Stephan_, Apr 29 2014]

%F 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

%F a(n) ~ (1/432)*n^3. - _Ralf Stephan_, Apr 29 2014

%F 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

%p 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

%t 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 *)

%o (MAGMA) MolienSeries(CoxeterGroup("F4")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

%o (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

%o (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

%o (Sage)

%o def A008670_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))).list()

%o A008670_list(70) # _G. C. Greubel_, Sep 08 2019

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

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)