OFFSET
0,3
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 linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).
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)
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 *)
PROG
(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
(Magma) R<x>:=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
(Sage)
def A097869_list(prec):
P.<x> = 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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 02 2004
STATUS
approved