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%I #19 Sep 08 2022 08:46:10
%S 1,-2,-3,8,-2,-6,14,-12,-9,20,-16,-12,31,-2,-15,32,-24,-24,38,-28,-21,
%T 44,-12,-24,57,-36,-27,72,-40,-30,62,-16,-42,68,-48,-36,74,-62,-48,80,
%U -2,-42,108,-60,-45,112,-64,-60,98,-24,-51,104,-96,-54,110,-76,-57
%N Expansion of q^(-1/2) * (eta(q) * eta(q^2) * eta(q^6) / eta(q^3))^2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A252651/b252651.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(-x^4)^4 * f(-x^6)^2 / f(x^2, x^4)^2 = f(-x^4)^4 * f(-x^2, -x^10)^2 / f(-x^12)^2 in powers of x where f() is a Ramanujan theta function.
%F Expansion of q^(-1/2) * sqrt(b(q) / (3 * c(q))) * b(q^2) * c(q^2) in powers of q where b(), c() are cubic AGM theta functions.
%F Euler transform of period 6 sequence [ -2, -4, 0, -4, -2, -4, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A118272.
%F G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 - x^k + x^(2*k))^2.
%F -2 * a(n) = A252650(2*n + 1).
%e G.f. = 1 - 2*x - 3*x^2 + 8*x^3 - 2*x^4 - 6*x^5 + 14*x^6 - 12*x^7 - 9*x^8 + ...
%e G.f. = q - 2*q^3 - 3*q^5 + 8*q^7 - 2*q^9 - 6*q^11 + 14*q^13 - 12*q^15 + ...
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/2)* (eta[q]*eta[q^2]*eta[q^6]/eta[q^3])^2, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Apr 07 2018 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A))^2, n))};
%o (Magma) A := Basis( ModularForms( Gamma0(36), 2), 115); A[2] - 2*A[4] - 3*A[6] + 8*A[8] - 2*A[10];
%Y Cf. A118272, A252650.
%K sign
%O 0,2
%A _Michael Somos_, Mar 22 2015
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