A097870 Molien series for group of order 4608 acting on joint weight enumerators of a pair of binary doubly-even self-dual codes.

1, 2, 4, 10, 17, 27, 45, 66, 92, 130, 173, 223, 289, 362, 444, 546, 657, 779, 925, 1082, 1252, 1450, 1661, 1887, 2145, 2418, 2708, 3034, 3377, 3739, 4141, 4562, 5004, 5490, 5997, 6527, 7105, 7706, 8332, 9010, 9713, 10443, 11229, 12042, 12884, 13786, 14717, 15679

Offset: 0

Author

N. J. A. Sloane, Sep 02 2004

Keywords

• nonn
• easy

Comments

This is the Molien series for the group of order 128 discussed in A097869 extended by the extra generator diag{1,1,i,i}. This group was not considered in the reference cited.

The first g.f. inserts zeros between each pair of terms; the second g.f. does not. - Colin Barker, Feb 12 2015

Links

• Colin Barker, 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.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A097869.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2 )); // G. C. Greubel, Feb 05 2020

Maple

m:=50; S:=series((1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2, x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 05 2020

Mathematica

CoefficientList[Series[(1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2, {x,0,50}], x] (* G. C. Greubel, Feb 05 2020 *)
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{1,2,4,10,17,27,45,66},50] (* Harvey P. Dale, Jun 11 2022 *)

PARI

Vec((x+1)*(x^2-x+1)*(x^4+x^3+x^2+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Feb 12 2015

Sage

def A097870_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2 ).list()
A097870_list(50) # G. C. Greubel, Feb 05 2020

Formula

G.f.: (1 + x^2 + 2*x^3 + x^4 + x^5 + x^6 + x^7)/(1 - 2*x + x^2 - 2*x^3 +

4*x^4 - 2*x^5 + x^6 - 2*x^7 + x^8).

G.f.: (1+x)*(1-x+x^2)*(1+x^2+x^3+x^4) / ((1-x)^4*(1+x+x^2)^2). - Colin Barker, Feb 12 2015