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%I #15 Jan 09 2017 04:19:07
%S 1,1,0,-2,-3,0,-4,-2,0,6,12,0,8,-4,0,4,-9,0,-16,6,0,-24,-12,0,7,8,0,8,
%T 3,0,44,4,0,18,12,0,-34,-16,0,-12,33,0,-40,-24,0,24,-60,0,-33,7,0,-16,
%U 27,0,72,8,0,-6,24,0,50,44,0,-8,-24,0,8,18,0,-24
%N Expansion of c(q) * b(q^2) * 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="/A208384/b208384.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)^2 * eta(q^3)^3 * eta(q^6)^2 / eta(q) in powers of q.
%F Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -6, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 8748^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A116418.
%F G.f.: x * Product_{k>0} (1 - x^(2*k))^2 * (1 - x^(3*k))^3 * (1 - x^(6*k))^2 / (1 - x^k).
%F a(3*n) = 0. a(3*n + 1) = A116418(n). a(3*n + 2) = A122407(n).
%e G.f. = q + q^2 - 2*q^4 - 3*q^5 - 4*q^7 - 2*q^8 + 6*q^10 + 12*q^11 + 8*q^13 + ...
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^3 QPochhammer[ q^6]^2 / QPochhammer[ q], {q, 0, n}]; (* _Michael Somos_, Mar 30 2015 *)
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^6 + A)^2 / eta(x + A), n))};
%Y Cf. A116418, A122407.
%K sign
%O 1,4
%A _Michael Somos_, Feb 25 2012
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