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%I #27 Mar 12 2021 22:24:46
%S 1,-12,-12,-12,-12,-72,-12,-96,-12,-12,-72,-144,-12,-168,-96,-72,-12,
%T -216,-12,-240,-72,-96,-144,-288,-12,-372,-168,-12,-96,-360,-72,-384,
%U -12,-144,-216,-576,-12,-456,-240,-168,-72,-504,-96,-528,-144,-72,-288
%N Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
%H G. C. Greubel, <a href="/A186100/b186100.txt">Table of n, a(n) for n = 0..1000</a>
%H J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
%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 b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
%F Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%F Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - _Michael Somos_, Jul 07 2015
%F a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
%F a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- _Michael Somos_, Jul 07 2015
%e G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...
%t a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* _Michael Somos_, Jul 07 2015 *)
%t a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* _Michael Somos_, Jul 07 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};
%o (PARI) {a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};
%Y Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.
%K sign
%O 0,2
%A _Michael Somos_, Feb 12 2011
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