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A028977
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Theta series of 8-d 6-modular lattice G_2 tensor F_4 (or A_2 tensor D_4) with det 1296 and minimal norm 4 in powers of q^2.
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2
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1, 0, 72, 192, 504, 576, 2280, 1728, 4248, 4800, 7920, 6336, 19416, 10368, 21312, 22464, 33624, 24192, 63048, 32832, 65808, 60864, 83232, 57600, 155640, 76032, 137520, 130944, 180288, 116928, 290736
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OFFSET
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0,3
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COMMENTS
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Proposition 7.6 [McKay and Sebbar, 2000, p. 272, equ. (7.8)] expresses the theta series as a Schwarzian of A007258 and tau. - Michael Somos, Jun 05 2015
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
G. Nebe and N. J. A. Sloane, Home page for this lattice
E. M. Rains and N. J. A. Sloane, The Shadow Theory of Modular and Unimodular Lattices, J. Number Theory, 73 (1998), 359-389.
Index entries for sequences related to D_4 lattice
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FORMULA
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Expansion of ((eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 - (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5)^2 - 8 * (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12))^2 in powers of q. - Michael Somos, May 27 2012
A212817(n) = a(n) + 8 * A030209(n). - Michael Somos, May 27 2012
G.f. A(x) = g1(x)^2 * (1 - 4*g2(x) - 16*g2(x)^3 + 16*g2(x)^4) where g1(x) = A033712(x) and g2(x) = A212770(x). - Michael Somos, Apr 19 2015
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EXAMPLE
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G.f. = 1 + 72*x^2 + 192*x^3 + 504*x^4 + 576*x^5 + 2280*x^6 + 1728*x^7 + ...
G.f. = 1 + 72*q^4 + 192*q^6 + 504*q^8 + 576*q^10 + 2280*q^12 + 1728*q^14 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ With[{e1 = QPochhammer[ x] QPochhammer[ x^6], e2 = QPochhammer[ x^2] QPochhammer[ x^3]}, (e2^7 / e1^5 - x e1^7 /e2^5)^2 - 8 x (e1 e2)^2], {x, 0, n}]; (* Michael Somos, Apr 19 2015 *)
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PROG
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(PARI) {a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^3 + A); A = eta(x + A) * eta(x^6 + A); polcoeff( (B^7 / A^5 - x * A^7 / B^5)^2 - 8 * x * (A * B)^2, n))}; /* Michael Somos, May 27 2012 */
(Magma) A := Basis( ModularForms( Gamma0(6), 4), 32); A[1] + 72*A[3] + 192*A[4] + 504*A[5]; /* Michael Somos, Aug 20 2014 */
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CROSSREFS
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Cf. A007258, A030209, A033712, A212770, A212817.
Sequence in context: A044785 A254437 A304376 * A033693 A250786 A302886
Adjacent sequences: A028974 A028975 A028976 * A028978 A028979 A028980
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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