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%I #28 Sep 08 2022 08:44:50
%S 1,0,72,192,504,576,2280,1728,4248,4800,7920,6336,19416,10368,21312,
%T 22464,33624,24192,63048,32832,65808,60864,83232,57600,155640,76032,
%U 137520,130944,180288,116928,290736
%N 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.
%C 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
%H G. C. Greubel, <a href="/A028977/b028977.txt">Table of n, a(n) for n = 0..1000</a>
%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F8.6.html">Home page for this lattice</a>
%H E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/shad.html">The Shadow Theory of Modular and Unimodular Lattices</a>, J. Number Theory, 73 (1998), 359-389.
%H <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>
%F 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
%F A212817(n) = a(n) + 8 * A030209(n). - _Michael Somos_, May 27 2012
%F 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
%e G.f. = 1 + 72*x^2 + 192*x^3 + 504*x^4 + 576*x^5 + 2280*x^6 + 1728*x^7 + ...
%e G.f. = 1 + 72*q^4 + 192*q^6 + 504*q^8 + 576*q^10 + 2280*q^12 + 1728*q^14 + ...
%t 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 *)
%o (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 */
%o (Magma) A := Basis( ModularForms( Gamma0(6), 4), 32); A[1] + 72*A[3] + 192*A[4] + 504*A[5]; /* _Michael Somos_, Aug 20 2014 */
%Y Cf. A007258, A030209, A033712, A212770, A212817.
%K nonn
%O 0,3
%A _N. J. A. Sloane_