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%I #14 Oct 19 2018 09:20:52
%S 1,-12,54,-84,-147,756,-756,-1212,3510,-2028,-3402,7992,-6132,-5964,
%T 18576,-10584,-14619,29484,-18252,-21084,55188,-28896,-35964,73008,
%U -49140,-46128,118692,-54516,-73896,146340,-95256,-92148,224694,-111888,-132678,260064
%N Expansion of s(q)^4 in powers of q where s() is a cubic AGM function.
%C Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).
%H Seiichi Manyama, <a href="/A133078/b133078.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%F Expansion of (eta(q)^3 / eta(q^3))^4 in powers of q.
%F Euler transform of period 3 sequence [ -12, -12, -8, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033690.
%F G.f.: ( Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k)) )^4.
%e G.f. = 1 - 12*q + 54*q^2 - 84*q^3 - 147*q^4 + 756*q^5 - 756*q^6 - 1212*q^7 + ...
%t QP = QPochhammer; A133078[n_] := SeriesCoefficient[(QP[q]^3/QP[q^3])^4, {q, 0, n}]; Table[A133078[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 20 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A)^3 / eta(x^3 + A) )^4, n))};
%Y Cf. A033690.
%K sign
%O 0,2
%A _Michael Somos_, Sep 08 2007
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