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%I #19 May 28 2023 20:47:07
%S 1,-3,3,5,-18,15,24,-75,57,86,-252,183,262,-744,522,725,-1998,1365,
%T 1852,-4986,3336,4436,-11736,7719,10103,-26322,17067,22040,-56682,
%U 36306,46336,-117867,74700,94378,-237744,149277,186926,-466836,290706,361126,-895014,553224
%N Expansion of (c(q^2)/c(q))^3 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 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C In the arXiv:2305.13951 paper on page 21 is this: "The q-expansion of y coincides with the sequence A123633 in the OEIS". - _Michael Somos_, May 26 2023
%H Seiichi Manyama, <a href="/A123633/b123633.txt">Table of n, a(n) for n = 1..10000</a>
%H Xuhang Jiang, Xing Wang, Li Lin Yang and Jing-Bang Zhao, <a href="https://arxiv.org/abs/2305.13951">Epsilon-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves</a>, arXiv:2305.13951 [hep-th], 2023. See page 21.
%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 / (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
%F Euler transform of period 6 sequence [ -3, 0, 6, 0, -3, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v)= u^2 - v - u*v * (6 + 8*v).
%F G.f.: x * (Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3 )^3.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128642.
%F A128636(n) = a(n) unless n = 0. Convolution inverse of A105559.
%F Convolution cube of A092848.
%F Convolution with A123330 is A093829. - _Michael Somos_, May 26 2023
%e G.f. = q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + 57*q^9 + ...
%t a[ n_] := SeriesCoefficient[ q / (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3, {q, 0, n}]; (* _Michael Somos_, Feb 19 2015 *)
%t a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 1, n, 2}] / Product[ 1 - q^k, {k, 3, n, 6}]^3)^3, {q, 0, n}]; (* _Michael Somos_, Feb 19 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3 * (eta(x^6 + A) / eta(x^3 + A))^9, n))};
%o (Magma) M := Basis(ModularForms(Gamma1(6), 1), 43); M1 := M[1]; M2 := M[2]; A<q> := M2/(M1 + 2*M2); A; /* _Michael Somos_, May 26 2023 */
%Y Cf. A092848, A093829, A105559, A123330, A128636, A128642.
%K sign
%O 1,2
%A _Michael Somos_, Oct 03 2006, Jan 21 2009
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