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%I #15 Mar 12 2021 22:24:45
%S 1,-2,-2,6,-2,-4,6,0,-2,-2,-4,0,6,-4,0,12,-2,-4,-2,0,-4,0,0,0,6,-6,-4,
%T 6,0,-4,12,0,-2,0,-4,0,-2,-4,0,12,-4,-4,0,0,0,-4,0,0,6,-2,-6,12,-4,-4,
%U 6,0,0,0,-4,0,12,-4,0,0,-2,-8,0,0,-4,0,0,0,-2,-4
%N Expansion of (3 * phi(q^3)^2 - phi(q)^2) / 2 in powers of q where phi() 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="/A138949/b138949.txt">Table of n, a(n) for n = 0..1000</a>
%H L.-C. Shen, <a href="https://doi.org/10.1090/S0002-9939-1994-1212287-3">On the Modular Equations of Degree 3</a>, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1108, Eq. (3.20).
%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 phi(-q) * phi(-q^2) * chi(q^3) / chi(-q^3) in powers of q where phi(), chi() are Ramanujan theta functions.
%F Expansion of eta(q)^2 * eta(q^2) * eta(q^6)^3 / (eta(q^3)^2 * eta(q^4) * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [ -2, -3, 0, -2, -2, -4, -2, -2, 0, -3, -2, -2, ...].
%F Moebius transform is period 12 sequence [ -2, 0, 8, 0, -2, 0, 2, 0, -8, 0, 2, 0, ...].
%F a(n) = -2 * b(n) where b() is multiplicative and b(2^e) = 1, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113446.
%F G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^2 / ((1 + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))).
%F G.f.: 1 - 2 * Sum_{k>0} (f(3*k - 2) + f(3*k - 1) - 2 * f(3*k)) where f(n) := x^n / (1 + x^(2*n)).
%F a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n).
%F a(n) = -2 * A138950(n) unless n=0. a(2*n + 1) = -2 * A116604(n).
%F a(3*n + 1) = A122865(n). a(3*n + 2) = -2 * A122856(n). a(4*n + 1) = -2 * A008441(n).
%e G.f. = 1 - 2*q - 2*q^2 + 6*q^3 - 2*q^4 - 4*q^5 + 6*q^6 - 2*q^8 - 2*q^9 + ...
%t a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^3]^2 - EllipticTheta[ 3, 0, q]^2) / 2, {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* _Michael Somos_, Sep 07 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, kronecker(-4, n/d) * [2, -1, -1][d%3 + 1]))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, -1 + 2 * (-1)^e, p%12 < 6, e+1, 1-e%2))) };
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^6 + A)^3 / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))};
%Y Cf. A008441, A116604, A122856, A122865, A138950.
%K sign
%O 0,2
%A _Michael Somos_, Apr 03 2008
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