A108094
Coefficients of series whose 16th power is the theta series of the 16-dimensional Barnes-Wall lattice (see A008409).
Original entry on oeis.org
1, 0, 270, 3840, -514080, -15413760, 1283087040, 62644907520, -3378279124350, -252933976704000, 8502815843769600, 1007506223570707200, -17757117956815481280, -3942183666885514421760, 14527133705347401150720, 15088544258811557869278720, 144818514010649047069497600
Offset: 0
More precisely, the theta series of the Barnes-Wall lattice begins 1 + 4320*q^2 + 61440*q^3 + 522720*q^4 + 2211840*q^5 + 8960640*q^6 + 23224320*q^7 + ... and the 16th root of this is 1 + 270*q^2 + 3840*q^3 - 514080*q^4 - 15413760*q^5 + 1283087040*q^6 + 62644907520*q^7 - ...
- Seiichi Manyama, Table of n, a(n) for n = 0..551
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
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f[q_] := 1/2 (EllipticTheta[2, 0, q]^16 + EllipticTheta[3, 0, q]^16 + EllipticTheta[4, 0, q]^16 + 30 EllipticTheta[2, 0, q]^8 EllipticTheta[3, 0, q]^8);
CoefficientList[f[q]^(1/16) + O[q]^17, q] (* Jean-François Alcover, Aug 17 2018 *)
A320729
Inverse Euler transform of A008409.
Original entry on oeis.org
0, 4320, 61440, -8810640, -263208960, 22737197280, 1104970014720, -61319781714960, -4559126871265280, 157589681950830816, 18460588655551795200, -337394373951169867920, -73200715363477357608960, 306095068890572836888800, 283363485933298319434493952, 2600519516007145850005282800
Offset: 1
1 + 4320*q^2 + 61440*q^3 + 522720*q^4 + ... = (1-q^2)^(-4320) * (1-q^3)^(-61440) * (1-q^4)^8810640 * ... .
A008774
Theta series of (probably nonexistent) exceptionally good 16-dimensional sphere packing.
Original entry on oeis.org
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
Offset: 0
1 + 7680*q^3 + 4320*q^4 + 276480*q^5 + 61440*q^6 + 2903040*q^7 + ...
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QP = QPochhammer; a[n_] := Module[{A, A1, A2, A4}, A = x*O[x]^n; A1 = QP[x+ A]^8; A2 = QP[x^2+A]^8; A4 = QP[x^4+A]^8; SeriesCoefficient[(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]]; a[0] = 1; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
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{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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