%I #17 Nov 23 2023 08:01:20

%S 1,-6,12,-6,-6,0,12,-12,12,-6,0,0,-6,-12,24,0,-6,0,12,-12,0,-12,0,0,

%T 12,-6,24,-6,-12,0,0,-12,12,0,0,0,-6,-12,24,-12,0,0,24,-12,0,0,0,0,-6,

%U -18,12,0,-12,0,12,0,24,-12,0,0,0,-12,24,-12,-6,0,0,-12,0,0,0,0,12,-12,24,-6,-12,0,24,-12,0,-6,0,0,-12,0,24

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

%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).

%H G. C. Greubel, <a href="/A122859/b122859.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>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F Expansion of 2*a(q^2) - a(q) = b(q)^2 / b(q^2) in powers of q where a(), b() are cubic AGM theta functions.

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

%F Euler transform of period 6 sequence [ -6, -3, -4, -3, -6, -2, ...].

%F Moebius transform is period 6 sequence [ -6, 18, 0, -18, 6, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v*(u+v)^2 - 2*u*w*(v+w).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1-u2-u3+u6) * (u1+2*u2+u3) - (2*u1+u2-2*u3-u6) * (u1+2*u2-u3).

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

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

%F / (1 + x^(3*k - 1))).

%F a(3*n) = a(4*n) = a(n). a(6*n + 5) = 0.

%F (-1)^n * a(n) = A113660(n). -6 * a(n) = A122860(n) if n>0.

%F a(2*n) = A227354(n). a(2*n + 1) = -6 * A033762(n). a(3*n + 1) = -6 * A033687(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 3) = -6 * A112605(n). a(6*n + 1) = -6 * A097195(n). a(8*n + 1) = -6 * A112606(n). a(8*n + 3) = -6 * A112608(n). a(8*n + 5) = -12 * A112607(n-1). a(8*n + 7) = -12 * A112609(n). a(12*n + 1) = -6 * A123884(n). a(12*n + 7) = -12 * A121361(n). - _Michael Somos_, Sep 27 2013

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - _Amiram Eldar_, Nov 23 2023

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

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3 / EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* _Michael Somos_, Sep 27 2013 *)

%o (PARI) {a(n)= if( n<1, n==0, 6 * sumdiv(n, d, (-1)^(n/d) * kronecker( -3, d)))}

%o (PARI) {a(n)= if( n<1, n==0, -6 * sumdiv(n, d, (2 + (-1)^d) * kronecker( -3, d)))}

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

%o (Sage) A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] - 6 *A[1] # _Michael Somos_, Sep 27 2013

%Y Cf. A033687, A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A113660, A121361, A122860, A123884, A227354.

%Y Cf. A000122, A000700, A010054, A121373.

%K sign

%O 0,2

%A _Michael Somos_, Sep 15 2006