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A157830 Coefficients of polynomial expansion of Golay C_24 enumeration Polynomial: p(x)=1 + 759*x^8 + 2576*x^12 + 759*x^16 + x^24; q(x)=x^24*(p(1/x). 0
1, 0, 0, 0, 0, 0, 0, 0, -759, 0, 0, 0, -2576, 0, 0, 0, 575322, 0, 0, 0, 3910368, 0, 0, 0, -429457542, 0, 0, 0, -4448043600, 0, 0, 0, 315448497771, 0, 0, 0, 4479379753856, 0, 0, 0, -227641291795533, 0, 0, 0, -4209068502252768, 0, 0, 0, 161001433246525844 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Conjecture:

An higher symmetry elliptical Invariant than E8 exists.

The Golay C_24 to C_12 symmetry seems to fit the bill!

Using the C_24 Golay enumeration polynomial

and the C_12 Golay enumeration polynomial:

s[x_]=(1 + 759*x^8 + 2576*x^12 + 759*x^16 + x^24)^3

/((24 + 440*x^3 + 264*x^6 + x^12)^3*(1 + 264 x^6 + 440 x^9 + 24 x^12)^3)

which is toral inversion symmetric:

s[1/x]=s[x]

which checks in Mathematica.

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, pp. 84-85.

LINKS

Table of n, a(n) for n=0..48.

FORMULA

p(x)=1 + 759*x^8 + 2576*x^12 + 759*x^16 + x^24;

q(x)=x^24*(p(1/x);

a(n)=coefficients(q(x)).

MATHEMATICA

f[x_] = 1 + 759*x^8 + 2576*x^12 + 759*x^16 + x^24;

g[x] = ExpandAll[x^24*f[1/x]];

a = Table[SeriesCoefficient[ Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]

CROSSREFS

Sequence in context: A221340 A290738 A104336 * A105547 A001293 A001380

Adjacent sequences:  A157827 A157828 A157829 * A157831 A157832 A157833

KEYWORD

sign

AUTHOR

Roger L. Bagula and Gary W. Adamson, Mar 07 2009

STATUS

approved

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Last modified September 25 11:15 EDT 2020. Contains 337337 sequences. (Running on oeis4.)