%I #14 Mar 12 2021 22:24:44

%S 1,-3,3,-3,3,0,3,-6,3,-3,0,0,3,-6,6,0,3,0,3,-6,0,-6,0,0,3,-3,6,-3,6,0,

%T 0,-6,3,0,0,0,3,-6,6,-6,0,0,6,-6,0,0,0,0,3,-9,3,0,6,0,3,0,6,-6,0,0,0,

%U -6,6,-6,3,0,0,-6,0,0,0,0,3,-6,6,-3,6,0,6,-6

%N Expansion of psi(-q)^3 / psi(-q^3) 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).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A132973/b132973.txt">Table of n, a(n) for n = 0..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 b(q^2)^2 / b(-q) = b(q) * b(q^4) / b(q^2) in powers of q where b() is a cubic AGM theta function.

%F Expansion of (a(q^2) + 2 * a(q^4) - a(q)) / 2 = (c(q)^2 - 5 * c(q) * c(q^4) + 4 * c(q^4)^2) / (3 * c(q^2)) in powers of q where a(), c() are cubic AGM theta functions. - _Michael Somos_, May 26 2013

%F Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3) * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [ -3, 0, -2, -3, -3, 0, -3, -3, -2, 0, -3, -2, ...].

%F Moebius transform is period 12 sequence [ -3, 6, 0, 0, 3, 0, -3, 0, 0, -6, 3, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A113447.

%F G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^3 / ((1 - x^(3*k)) * (1 + x^(6*k))).

%F G.f.: 1 + 3 * Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).

%F G.f.: 1 + 3 * ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).

%F a(n) = (-1)^n * A107760(n). Convolution inverse of A132974.

%F a(2*n) = A107760(n). a(2*n + 1) = -3 * A033762(n). a(3*n) = A132973(n). a(3*n + 1) = -3 * A227696(n). - _Michael Somos_, Oct 31 2015

%F a(6*n + 1) = -3 * A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. - _Michael Somos_, Oct 31 2015

%e G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*q^12 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]/2, {q, 0, n}]; (* _Michael Somos_, May 26 2013 *)

%o (PARI) {a(n) = if( n<1, n==0, 3 * (-1)^n * sumdiv(n, d, kronecker(-12, d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A )), n))};

%Y Cf. A033687, A097195, A107760, A132973, A132974, A227696.

%K sign

%O 0,2

%A _Michael Somos_, Sep 07 2007