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Search: id:A008774
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| A008774 |
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Theta series of (probably nonexistent) exceptionally good 16-dimensional sphere packing. |
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3
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| 1, 0, 0, 7680, 4320, 276480, 61440, 2903040, 522720, 16896000, 2211840, 68774400, 8960640, 221460480, 23224320, 603325440, 67154400, 1448202240, 135168000, 3154982400, 319809600, 6359654400, 550195200, 12016788480, 1147643520
(list; graph; listen)
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OFFSET
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0,4
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 190, Equation (47).
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FORMULA
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Expansion of ( E_4(q) * 2 * (E_4(q^2) - E_4(q^4)) + E_4(q^2) * (32 * E_4(q^4) - 17 * E_4(q^2)) ) / 15 in powers of q. - Michael Somos Nov 29 2007
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EXAMPLE
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1 + 7680*q^3 + 4320*q^4 + 276480*q^5 + 61440*q^6 + 2903040*q^7 + ...
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PROGRAM
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(PARI) {a(n) = local(A, A1, A2, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^8; A4 = eta(x^4 + A)^8; polcoeff( ( A1 * (A2^6 + x^2 * 32 * A2^3 * A4^3 + x^4 * 4096 * A4^6) + x^3 * 3840 * A4^4 * ( A1^2 * A4 + A2^3 ) ) / (A1 * A2^2 * A4^2 ), n))} /* Michael Somos Nov 29 2007 */
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CROSSREFS
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A008409(n) = a(2*n). 7680 * A135828(n) = a(2*n+3).
Adjacent sequences: A008771 A008772 A008773 this_sequence A008775 A008776 A008777
Sequence in context: A031854 A068245 A038011 this_sequence A076339 A105132 A133859
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KEYWORD
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nonn
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AUTHOR
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njas
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