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A231962 Expansion of b(q)^3 - (1/3)*c(q)^3 in powers of q where b(), c() are cubic AGM theta functions.

%I A231962 #9 Aug 08 2018 22:54:32
%S A231962 1,-18,0,-90,-234,0,-216,-900,0,-738,-1296,0,-1170,-3060,0,-1728,
%T A231962 -3690,0,-2160,-6516,0,-4500,-6480,0,-3672,-10818,0,-6570,-11700,0,
%U A231962 -6480,-17316,0,-8640,-15552,0,-9594,-24660,0,-15300,-22032,0,-10800,-33300,0,-17280
%N A231962 Expansion of b(q)^3 - (1/3)*c(q)^3 in powers of q where b(), c() are cubic AGM theta functions.
%C A231962 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A231962 G. C. Greubel, <a href="/A231962/b231962.txt">Table of n, a(n) for n = 0..2500</a>
%F A231962 Expansion of (eta(q)^3 / eta(q^3))^3 - 9 * q * (eta(q^3)^3 / eta(q))^3 in powers of q.
%F A231962 G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = - 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231961.
%F A231962 a(3*n + 2) = 0. a(3*n + 1) = -18 * A231947(n). a(3*n) = A231961(n).
%e A231962 G.f. = 1 - 18*q - 90*q^3 - 234*q^4 - 216*q^6 - 900*q^7 - 738*q^9 + ...
%t A231962 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3/ eta[q^3])^3 - 9*(eta[q^3]^3/eta[q])^3, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 08 2018 *)
%o A231962 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 - 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}
%Y A231962 Cf. A231947, A231961.
%K A231962 sign
%O A231962 0,2
%A A231962 _Michael Somos_, Nov 15 2013