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%I #16 Sep 08 2022 08:46:02
%S 1,8,56,168,536,624,2328,1600,4184,4872,7824,6432,19320,10672,21568,
%T 22320,33752,23184,62904,32992,66000,61248,83040,58944,155832,75320,
%U 136912,130728,179776,117168,291024,142720,269528,236448,307440,207744,528024,243952
%N Theta series of direct sum of 2 copies of 4-dimensional lattice QQF.4.i.
%H G. C. Greubel, <a href="/A212817/b212817.txt">Table of n, a(n) for n = 0..1000</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 in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 576 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F Convolution square of A125514.
%F a(n) = A028977(n) + 8 * A030209(n). - _Michael Somos_, Jun 05 2015
%e G.f. = 1 + 8*x + 56*x^2 + 168*x^3 + 536*x^4 + 624*x^5 + 2328*x^6 + 1600*x^7 + ...
%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 ], {x, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)
%o (PARI) {a(n) = my(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, n))};
%o (PARI) {a(n) = my(G); if( n<0, 0, G = [ 2, 0, 1, 1; 0, 2, 1, 1; 1, 1, 4, 1; 1, 1, 1, 4 ]; polcoeff( (1 + 2 * x * Ser( qfrep( G, n, 1)))^2, n))};
%o (Magma) A := Basis( ModularForms( Gamma0(6), 4), 38); A[1] + 8*A[2] + 56*A[3] + 168*A[4] + 536*A[5]; /* _Michael Somos_, Jun 04 2015 */
%Y Cf. A028977, A030209, A125514.
%K nonn
%O 0,2
%A _Michael Somos_, May 27 2012
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