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%I #15 Sep 08 2022 08:46:08
%S 1,-6,9,14,-54,36,65,-162,126,148,-438,252,344,-756,513,546,-1458,756,
%T 1022,-2064,1332,1352,-3510,1764,2198,-4374,2808,2710,-6804,3276,4161,
%U -7992,4914,4816,-11826,5616,6860,-13188,8190,7658,-18576,8892,10804,-20412
%N Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C In McKay and Sebbar on page 274 in equation (8.2) the last term on the right side is a multiple of the g.f.
%H G. C. Greubel, <a href="/A242042/b242042.txt">Table of n, a(n) for n = 2..2500</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), no. 2, 255-275. MR1795562 (2001m:11063)
%F Expansion of (eta(q) * eta(q^9))^6 / eta(q^3)^4 in powers of q.
%F Euler transform of period 9 sequence [ -6, -6, -2, -6, -6, -2, -6, -6, -8, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F G.f.: x^2 * Product_{k>0} (1 - x^k)^6 * (1 - x^(9*k))^6 / (1 - x^(3*k))^4.
%F Convolution square of A106401.
%F a(3*n) = -6 * A198956(n). a(3*n + 1) = 9 * A033690(n).
%e G.f. = q^2 - 6*q^3 + 9*q^4 + 14*q^5 - 54*q^6 + 36*q^7 + 65*q^8 - 162*q^9 + ...
%t a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q] QPochhammer[ q^9])^6 / QPochhammer[ q^3]^4, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^6 / eta(x^3 + A)^4, n))};
%o (Magma) A := Basis( ModularForms( Gamma0(9), 4), 19); A[3] - 6*A[4] + 9*A[5];
%Y Cf. A033680, A106401, A198956.
%K sign
%O 2,2
%A _Michael Somos_, Aug 12 2014
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