A092544 Molien series for complete weight enumerators of self-dual codes over Z/8Z.

1, 1, 1, 5, 23, 54, 120, 263, 557, 1058, 1883, 3260, 5490, 8844, 13754, 20916, 31201, 45447, 64787, 90925, 125881, 171626, 230670, 306511, 403205, 524760, 676017, 863456, 1094420, 1376184, 1717308, 2128800, 2622977, 3212093, 3910205, 4734941, 5705771, 6842078

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..10000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Index entries for Molien series

Formula

G.f.: u1/u2, where u1 = subs(x -> x^2, f), u2 = (1-x^2)^4*(1-x^8)^2*(1-x^16)^2, and f = (1+x)^2*(1 -5*x +12*x^2 -16*x^3 +24*x^4 -37*x^5 +67*x^6 -83*x^7 +118*x^8 -146*x^9 +186*x^10 -158*x^11 +137*x^12 -111*x^13 +113*x^14 -81*x^15 + 69*x^16 -53*x^17 +38*x^18 -14*x^19 +3*x^20), even terms only.

G.f.: (1+x)^2*(1 -5*x +12*x^2 -16*x^3 +24*x^4 -37*x^5 +67*x^6 -83*x^7 +118*x^8 -146*x^9 +186*x^10 -158*x^11 +137*x^12 -111*x^13 +113*x^14 -81*x^15 + 69*x^16 -53*x^17 +38*x^18 -14*x^19 +3*x^20)/((1-x)^4*(1-x^4)^2*(1-x^8)^2). - G. C. Greubel, Feb 01 2020

Maple

g:= (1-3*x+3*x^2+3*x^3+4*x^4-5*x^5+17*x^6+14*x^7+19*x^8+7*x^9 +12*x^10 +68*x^11 +7*x^12+5*x^13+28*x^14+34*x^15+20*x^16+4*x^17+x^18 +9*x^19 +13*x^20 -8*x^21 +3*x^22)/((1-x)^4*(1-x^4)^2*(1-x^8)^2):
S:= series(g, x, 100):
seq(coeff(S, x, j), j=0..100); # Robert Israel, Mar 01 2016

Mathematica

CoefficientList[Series[(1+x)^2*(1 -5*x +12*x^2 -16*x^3 +24*x^4 -37*x^5 +67*x^6 -83*x^7 +118*x^8 -146*x^9 +186*x^10 -158*x^11 +137*x^12 -111*x^13 +113*x^14 -81*x^15 + 69*x^16 -53*x^17 +38*x^18 -14*x^19 +3*x^20)/((1-x)^4*(1-x^4)^2*(1-x^8)^2), {x, 0, 40}], x] (* G. C. Greubel, Feb 01 2020 *)

Prog

(Sage)
def g(x): return (1+x)^2*(1 -5*x +12*x^2 -16*x^3 +24*x^4 -37*x^5 +67*x^6 -83*x^7 +118*x^8 -146*x^9 +186*x^10 -158*x^11 +137*x^12 -111*x^13 +113*x^14 -81*x^15 + 69*x^16 -53*x^17 +38*x^18 -14*x^19 +3*x^20)/((1-x)^4*(1-x^4)^2*(1-x^8)^2)
[( g(x) ).series(x, n+1).list()[n] for n in (0..40)] # G. C. Greubel, Feb 01 2020

Crossrefs

Cf. A092545, A092546, A092547.

Sequence in context: A019267 A053664 A186030 * A319087 A098499 A075565

Adjacent sequences: A092541 A092542 A092543 * A092545 A092546 A092547

Keyword

nonn

Author

N. J. A. Sloane, Apr 09 2004

Status

approved