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<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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved