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Search: id:A157830
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| A157830 |
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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). |
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| 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; listen)
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OFFSET
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0,9
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COMMENT
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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.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, pp. 84-85.
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FORMULA
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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)).
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MATHEMATICA
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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}]
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CROSSREFS
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Sequence in context: A101833 A068683 A104336 this_sequence A105547 A001293 A001380
Adjacent sequences: A157827 A157828 A157829 this_sequence A157831 A157832 A157833
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KEYWORD
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sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Mar 07 2009
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