A097869 Expansion of g.f.: (1+x^4+x^5+x^9)/((1-x)*(1-x^2)*(1-x^4)^2).

1, 1, 2, 2, 6, 7, 11, 12, 21, 25, 34, 38, 54, 63, 79, 88, 113, 129, 154, 170, 206, 231, 267, 292, 341, 377, 426, 462, 526, 575, 639, 688, 769, 833, 914, 978, 1078, 1159, 1259, 1340, 1461, 1561, 1682, 1782, 1926, 2047, 2191, 2312, 2481, 2625, 2794, 2938, 3134, 3303

Offset: 0

Author

N. J. A. Sloane, Sep 02 2004

Keywords

• nonn

Comments

Molien series for group of order 128 acting on joint weight enumerators of a pair of binary self-dual codes is (1+x^8+x^10+x^18)/((1-x^2)*(1-x^4)*(1-x^8)^2).

Links

• G. C. Greubel, 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. 804, Sect. 5.4.1.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Crossrefs

Cf. A097870.

Programs

Magma

R:=PowerSeriesRing(Integers(), 55); Coefficients(R!( (1+x^4)*(1+x^5)/((1-x)*(1-x^2)*(1-x^4)^2) )); // G. C. Greubel, Feb 05 2020

Maple

m:=55; S:=series((1+x^4)*(1+x^5)/((1-x)*(1-x^2)*(1-x^4)^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 05 2020

Mathematica

CoefficientList[Series[(1+x^4)*(1+x^5)/((1-x)*(1-x^2)*(1-x^4)^2), {x,0,55}], x] (* G. C. Greubel, Feb 05 2020 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1},{1,1,2,2,6,7,11,12,21,25},60] (* Harvey P. Dale, Jun 19 2021 *)

PARI

Vec( (1+x^4)*(1+x^5)/((1-x)*(1-x^2)*(1-x^4)^2) +O('x^55) ) \\ G. C. Greubel, Feb 05 2020

Sage

def A097869_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^4)*(1+x^5)/((1-x)*(1-x^2)*(1-x^4)^2) ).list()
A097869_list(55) # G. C. Greubel, Feb 05 2020

Formula

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

From Greg Dresden, Jun 22 2021: (Start)

a(2*n) = (1/48)*(30 + 18*(-1)^n + 64*n + 12*n^2 + 8*n^3),

a(2*n+1) = (1/48)*(36 + 12*(-1)^n + 16*n + 8*n^2)*(1 + n). (End)