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A181976 Expansion of a(q) * b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.

%I A181976 #9 Aug 12 2018 01:00:31
%S A181976 1,0,-27,72,0,-216,270,0,-459,720,0,-1080,936,0,-1350,2160,0,-2592,
%T A181976 2214,0,-2808,3600,0,-4752,4590,0,-4590,6552,0,-7560,5184,0,-7371,
%U A181976 10800,0,-10800,9360,0,-9774,12240,0,-15120,13500,0,-14040,17712,0,-19872,14760
%N A181976 Expansion of a(q) * b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.
%C A181976 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A181976 G. C. Greubel, <a href="/A181976/b181976.txt">Table of n, a(n) for n = 0..2500</a>
%F A181976 Expansion of b(q^3)^3 - 3 * b(q) * c(q^3)^2 in powers of q where b(), c() are cubic AGM theta functions.
%F A181976 Expansion of b(q^3)^2 * (b(q) + c(q^3)) in powers of q^3 where b(), c() are cubic AGM theta functions.
%F A181976 Expansion of (eta(q)^9 + 9 * q * eta(q)^6 * eta(q^9)^3) / eta(q^3)^3 in powers of q.
%F A181976 a(3*n + 1) = 0. a(3*n) = A004007(n).
%e A181976 G.f. = 1 - 27*q^2 + 72*q^3 - 216*q^5 + 270*q^6 - 459*q^8 + 720*q^9 + ...
%t A181976 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^9 + 9*q*eta[q]^6*eta[q^9]^3)/eta[q^3]^3, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 11 2018 *)
%o A181976 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^9 + 9 * x * eta(x + A)^6 * eta(x^9 + A)^3) / eta(x^3 + A)^3, n))};
%Y A181976 Cf. A004007.
%K A181976 sign
%O A181976 0,3
%A A181976 _Michael Somos_, Apr 04 2012