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A208589 Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A208589 #17 Feb 16 2025 08:33:16
%S A208589 1,2,0,0,1,-2,0,0,-1,4,0,0,0,-6,0,0,1,8,0,0,0,-12,0,0,-1,18,0,0,-1,
%T A208589 -24,0,0,2,32,0,0,1,-44,0,0,-2,58,0,0,-1,-76,0,0,2,100,0,0,1,-128,0,0,
%U A208589 -3,164,0,0,-1,-210,0,0,4,264,0,0,2,-332,0,0,-5,416
%N A208589 Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A208589 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208589 G. C. Greubel, <a href="/A208589/b208589.txt">Table of n, a(n) for n = 0..2500</a>
%H A208589 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208589 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208589 Expansion of q^(1/2) * eta(q^2)^5 / (eta(q)^2 * eta(q^4) * eta(q^8)^2) in powers of q.
%F A208589 Given g.f. A(x), then B(q) = (A(q^2) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - (v - 4) * (u - 4)^2.
%F A208589 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 + u) - 3*u*v * (2*(u^2 + v^2) - 11). - _Michael Somos_, Jul 05 2014
%F A208589 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208850. - _Michael Somos_, Jul 05 2014
%F A208589 a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = 2 * A083365(n).
%F A208589 Convolution square is A131125. Convolution inverse is A210063. - _Michael Somos_, Jul 05 2014
%e A208589 G.f. = 1 + 2*x + x^4 - 2*x^5 - x^8 + 4*x^9 - 6*x^13 + x^16 + 8*x^17 - 12*x^21 + ...
%e A208589 G.f. = 1/q + 2*q + q^7 - 2*q^9 - q^15 + 4*q^17 - 6*q^25 + q^31 + 8*q^33 + ...
%t A208589 a[ n_] := SeriesCoefficient[ 2 q^(1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2], {q, 0, n}]; (* _Michael Somos_, Jul 05 2014 *)
%o A208589 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2), n))};
%Y A208589 Cf. A029838, A083365, A131125. A208850, A210063.
%K A208589 sign
%O A208589 0,2
%A A208589 _Michael Somos_, Feb 29 2012