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

0,2

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).

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

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 *)

Prog

(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
(Magma) R<x>:=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
(Sage)
def A097870_list(prec):
    P.<x> = 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

Crossrefs

Cf. A097869.

Sequence in context: A264585 A077635 A125754 * A244474 A301739 A152231

Adjacent sequences: A097867 A097868 A097869 * A097871 A097872 A097873

Keyword

nonn,easy

Author

N. J. A. Sloane, Sep 02 2004

Status

approved