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A259657 Expansion of phi(-x^3) * f(-x^4)^3 / f(-x^12) in powers of x where phi(), f() are Ramanujan theta functions.

%I A259657 #10 Feb 16 2025 08:33:26
%S A259657 1,0,0,-2,-3,0,0,6,0,0,0,0,8,0,0,-12,-9,0,0,6,0,0,0,0,12,0,0,-2,-12,0,
%T A259657 0,18,0,0,0,0,6,0,0,-24,-12,0,0,6,0,0,0,0,20,0,0,-12,-12,0,0,24,0,0,0,
%U A259657 0,24,0,0,-12,-21,0,0,6,0,0,0,0,0,0,0,-14,-24
%N A259657 Expansion of phi(-x^3) * f(-x^4)^3 / f(-x^12) in powers of x where phi(), f() are Ramanujan theta functions.
%C A259657 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259657 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A259657 G. C. Greubel, <a href="/A259657/b259657.txt">Table of n, a(n) for n = 0..1000</a>
%H A259657 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259657 Expansion of phi(-x^3) * b(x^4) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
%F A259657 Expansion of eta(q^3)^2 * eta(q^4)^3 / (eta(q^6) * eta(q^12)) in powers of q.
%F A259657 Euler transform of period 12 sequence [ 0, 0, -2, -3, 0, -1, 0, -3, -2, 0, 0, -3, ...].
%F A259657 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 72^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259655.
%F A259657 G.f.: Product_{k>0} (1 - x^(4*k))^3 / ((1 + x^(3*k))^2 * (1 + x^(6*k))).
%F A259657 a(3*n + 1) = -3 * A143161(n-1). a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(12*n) = A014453(n).
%e A259657 G.f. = 1 - 2*x^3 - 3*x^4 + 6*x^7 + 8*x^12 - 12*x^15 - 9*x^16 + 6*x^19 + ...
%t A259657 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^4]^3 / QPochhammer[ x^12], {x, 0, n}];
%t A259657 a[ n_] := SeriesCoefficient[ QPochhammer[x^4]^3 / (QPochhammer[ -x^3, x^3]^2 QPochhammer[ -x^6, x^6]), {x, 0, n}];
%o A259657 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^3 / (eta(x^6 + A) * eta(x^12 + A)), n))};
%Y A259657 Cf. A014453, A143161, A259655.
%K A259657 sign
%O A259657 0,4
%A A259657 _Michael Somos_, Jul 02 2015