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%I #18 Mar 14 2023 09:34:13
%S 1,-2,3,0,-1,0,7,-8,6,0,1,0,8,-12,15,0,-7,0,18,-16,12,0,5,0,14,-26,24,
%T 0,-6,0,31,-24,18,0,-5,0,20,-28,42,0,-8,0,36,-48,24,0,13,0,31,-36,42,
%U 0,-25,0,56,-40,30,0,6,0,32,-64,63,0,-12,0,54,-48,48,0
%N Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A229615/b229615.txt">Table of n, a(n) for n = 2..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 (a(q) - a(q^2))^2 / 36 = c(q^2)^4 / (9 * c(q)^2) in powers of q where a(), c() are cubic AGM theta functions.
%F Expansion of ((eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3))^2 in powers of q.
%F Euler transform of period 6 sequence [ -2, 2, 4, 2, -2, -4, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/12) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229616.
%F G.f.: sum_{k>0} x^(6*k-4) / (1 - x^(6*k-4))^2 - 2 * x^(6*k-3) / (1 - x^(6*k-3))^2 + x^(6*k-2) / (1 - x^(6*k-2))^2.
%F G.f.: sum_{k>0} (3*k-2) * x^(6*k-4) / (1 - x^(6*k-4)) - (4*k-2) * x^(6*k-3) / (1 - x^(6*k-3)) + (3*k-1) * x^(6*k-2) / (1 - x^(6*k-2)).
%F a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 2) = A144614(n). a(6*n + 3) = -2 * A008438(n). a(6*n + 4) = 3 * A033686(n).
%F Convolution square of A093829.
%e G.f. = q^2 - 2*q^3 + 3*q^4 - q^6 + 7*q^8 - 8*q^9 + 6*q^10 + q^12 + ...
%t a[ n_] := If[n < 1, 0, Sum[ {0, 1, -2, 1, 0, 0}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
%t a[ n_] := If[n < 1, 0, Sum[ {0, 1/2, -2/3, 1/2, 0, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(3/2)]^6 / EllipticTheta[ 2, 0, q^(1/2)]^2 / 16, {q, 0, n}];
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * [0, 0, 1, -2, 1, 0][d%6 + 1]))};
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 1/2, -2/3, 1/2, 0][d%6 + 1]))};
%o (PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3))^2, n))};
%o (Sage) ModularForms( Gamma0(6), 2, prec=70).2;
%o (Magma) Basis( ModularForms( Gamma0(6), 2), 70)[3] /* _Michael Somos_, Mar 05 2023 */
%Y Cf. A008438, A033686, A093829, A144614, A229616.
%K sign
%O 2,2
%A _Michael Somos_, Sep 26 2013
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