A008796 Molien series for 3-dimensional group [2,3]+ = 223; also for group H_{1,2} of order 384.

1, 0, 2, 1, 4, 2, 7, 4, 10, 7, 14, 10, 19, 14, 24, 19, 30, 24, 37, 30, 44, 37, 52, 44, 61, 52, 70, 61, 80, 70, 91, 80, 102, 91, 114, 102, 127, 114, 140, 127, 154, 140, 169, 154, 184, 169, 200, 184, 217, 200, 234, 217, 252, 234, 271, 252, 290, 271, 310, 290, 331, 310, 352, 331, 374, 352, 397

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II Codes, Even Unimodular Lattices and Invariant Rings, IEEE Trans. Information Theory, Volume 45, Number 4, 1999, 1194-1205.
• Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).
• Index entries for Molien series

Crossrefs

Cf. A008795.

Programs

GAP

a:=[1,0,2,1,4,2,7];; for n in [8..70] do a[n]:=2*a[n-2]+a[n-3]-a[n-4]-2*a[n-5]+a[n-7]; od; a; # G. C. Greubel, Sep 11 2019

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^4)/((1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 11 2019

Maple

seq(coeff(series((1+x^4)/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 11 2019

Mathematica

LinearRecurrence[{0,2,1,-1,-2,0,1},{1,0,2,1,4,2,7},70] (* Harvey P. Dale, Apr 27 2014 *)
CoefficientList[Series[(1+x^4)/((1-x^2)^2*(1-x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Apr 28 2014 *)

PARI

a(n)=(9*(-1)^n*(2*n + 3) + 6*n^2 + 18*n + 24*!(n%3) + 21)/72 \\ Charles R Greathouse IV, Feb 10 2017

Sage

def A008796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^4)/((1-x^2)^2*(1-x^3))).list()
A008796_list(70) # G. C. Greubel, Sep 11 2019

Formula

G.f.: (1+x^4)/((1-x^2)^2*(1-x^3)).

a(n) = (1/72) * (9*(-1)^n*(2*n + 3) + 6*n^2 + 18*n + 29 - 8*A061347[n]). - Ralf Stephan, Apr 28 2014

Extensions

Definition clarified by N. J. A. Sloane, Feb 02 2018

More terms added by G. C. Greubel, Sep 11 2019