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%I #19 Aug 11 2018 18:17:46
%S 1,-2,0,-2,6,0,-4,4,0,6,-24,0,8,8,0,4,18,0,-16,-12,0,-24,24,0,7,-16,0,
%T 8,-6,0,44,-8,0,18,-24,0,-34,32,0,-12,-66,0,-40,48,0,24,120,0,-33,-14,
%U 0,-16,-54,0,72,-16,0,-6,-48,0,50,-88,0,-8,48,0,8,-36
%N Expansion of b(q) * c(q) * c(q^2) / 9 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).
%H Seiichi Manyama, <a href="/A208385/b208385.txt">Table of n, a(n) for n = 1..1000</a>
%H N. Elkies, <a href="http://mathoverflow.net/questions/84443/">Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418</a>
%F Expansion of eta(q)^2 * eta(q^3)^2 * eta(q^6)^3 / eta(q^2) in powers of q.
%F Euler transform of period 6 sequence [-2, -1, -4, -1, -2, -6, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 34992^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A122407.
%F G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(6*k))^3 / (1 - x^(2*k)).
%F a(3*n) = 0. a(3*n + 1) = A116418(n). a(3*n + 2) = -2 * A122407(n).
%e G.f. = q - 2*q^2 - 2*q^4 + 6*q^5 - 4*q^7 + 4*q^8 + 6*q^10 - 24*q^11 + 8*q^13 + ...
%t eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[eta[q]^2 *eta[q^3]^2*eta[q^6]^3/eta[q^2], {q, 0, 50}], q]] (* _G. C. Greubel_, Aug 11 2018 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^2 * eta(x^6 + A)^3 / eta(x^2 + A), n))};
%Y Cf. A116418, A122407.
%K sign
%O 1,2
%A _Michael Somos_, Feb 25 2012
%E a(40) corrected by _Seiichi Manyama_, Jan 09 2017
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