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 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 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:=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. = 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 STATUS approved

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