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%I #19 Mar 02 2021 06:28:34
%S 1,-1,-1,3,-2,-3,8,-5,-7,18,-12,-15,38,-24,-30,75,-46,-57,140,-86,
%T -104,252,-152,-183,439,-262,-313,744,-442,-522,1232,-725,-852,1998,
%U -1168,-1365,3182,-1852,-2150,4986,-2886,-3336,7700,-4436,-5106,11736,-6736,-7719,17673,-10103,-11538,26322
%N Expansion of c(q^4) / c(q) in powers of q where c() is a cubic AGM theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 7/4 + (3/4)*sqrt(3) - (1/4)*sqrt(72 + 42*sqrt(3)). - _Simon Plouffe_, Mar 01 2021
%H G. C. Greubel, <a href="/A123649/b123649.txt">Table of n, a(n) for n = 1..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 q * psi(q^6) * f(q, q^5) / f(q, q^2)^2 = q * f(-q^2, -q^10)^2 / (phi(-q^3) * f(q, q^5)) in powers of q where psi(), phi(), f() are Ramanujan theta functions. - _Michael Somos_, Jul 23 2013
%F Expansion of q * psi(-q) * psi(q^6)^3 / (psi(q^2) * psi(-q^3)^3) in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Jul 23 2013
%F Expansion of eta(q) * eta(q^12)^3 / (eta(q^4) * eta(q^3)^3) in powers of q.
%F Euler transform of period 12 sequence [ -1, -1, 2, 0, -1, 2, -1, 0, 2, -1, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 - 2*u) * (1 - 2*v).
%e q - q^2 - q^3 + 3*q^4 - 2*q^5 - 3*q^6 + 8*q^7 - 5*q^8 - 7*q^9 + ...
%t A123649[n_] := SeriesCoefficient[q*QPochhammer[q] *QPochhammer[q^12]^3/ (QPochhammer[q^4]* QPochhammer[q^3]^3), {q, 0, n}]; Rest[Table[ A123649[n], {n, 0, 50}]] (* _G. C. Greubel_, Oct 17 2017 *)
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^3 / eta(x^3 + A)^3 / eta(x^4 + A), n))}
%K sign
%O 1,4
%A _Michael Somos_, Oct 04 2006
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