A008718 Expansion of g.f.: (1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12)).

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 9, 10, 11, 12, 15, 16, 19, 20, 23, 26, 29, 30, 36, 39, 42, 45, 51, 54, 60, 63, 69, 75, 81, 84, 94, 100, 106, 112, 122, 128, 138, 144, 154, 164, 174, 180, 195, 205, 215, 225, 240, 250, 265, 275, 290, 305, 320, 330, 351, 366

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Molien series for genus-2 weight enumerators of binary self-dual codes is (1+x^18)/((1-x^2)*(1-x^8)*(1-x^12)*(1-x^24)). Exponents have been divided by 2 to get the sequence.

Or, Molien series for 4-dimensional representation of 2.{3,4,3}. This is the real 4-dimensional Clifford group of genus 2 and order 2304.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
• 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,-1,0,-1,1,0,0,0,1,-1,0,-1,0,1,0,1,-1).

Crossrefs

Cf. A008621, A024186.

Programs

Magma

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12)) )); // G. C. Greubel, Sep 09 2019

Maple

(1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12)); seq(coeff(series(%, x, n+1), x, n), n = 0..65); # modified by G. C. Greubel, Sep 09 2019

Mathematica

CoefficientList[Series[(1+x^9)/((1-x)(1-x^4)(1-x^6)(1-x^12)), {x,0,65}], x] (* Harvey P. Dale, Apr 01 2011 *)
LinearRecurrence[{1,0,1,0,-1,0,-1,1,0,0,0,1,-1,0,-1,0,1,0,1,-1}, {1,1,1, 1,2,2,3,3,4,5,6,6,9,10,11,12,15,16,19,20}, 65] (* Ray Chandler, Jul 16 2015 *)

PARI

my(x='x+O('x^65)); Vec((1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12))) \\ G. C. Greubel, Sep 09 2019

Sage

def A008718_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12))).list()
A008718_list(65) # G. C. Greubel, Sep 09 2019

Formula

a(n) ~ (1/864)*n^3. - Ralf Stephan, Apr 29 2014

G.f.: ( 1-x^3+x^6 ) / ( (1-x+x^2) *(x^4-x^2+1) *(1+x)^2 *(x^2+1)^2 *(1+x+x^2)^2 *(x-1)^4 ). - R. J. Mathar, Dec 18 2014