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%I #11 Mar 12 2021 22:24:46
%S 1,-1,-1,1,-1,1,1,-2,0,2,0,-1,1,-2,-1,5,-3,-2,5,-4,1,4,-7,0,6,-3,-3,6,
%T -6,-2,15,-12,-6,15,-12,3,15,-20,-2,20,-11,-7,19,-20,-7,40,-29,-14,40,
%U -34,3,40,-48,-5,52,-33,-17,52,-50,-14,93,-74,-32,97,-80,3,99,-112,-15
%N Expansion of f(-x) * f(-x^15) / (f(-x^3) * f(-x^5)) in powers of x where f() 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="/A193779/b193779.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 q^(-1/3) * eta(q) * eta(q^15) / (eta(q^3) * eta(q^5)) in powers of q.
%F Euler transform of period 15 sequence [ -1, -1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, ...].
%F Given g.f. A(x), then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (v - u^2) * (u - v^2) - 2 * u^2 * v^2.
%F A058685(3*n + 1) = - a(n). Convolution inverse of A058686.
%e 1 - x - x^2 + x^3 - x^4 + x^5 + x^6 - 2*x^7 + 2*x^9 - x^11 + x^12 + ...
%e q - q^4 - q^7 + q^10 - q^13 + q^16 + q^19 - 2*q^22 + 2*q^28 - q^34 + q^37 + ...
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/3)* eta[q]*eta[q^15]/(eta[q^3]*eta[q^5]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Apr 03 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^15 + A) / (eta(x^3 + A) * eta(x^5 + A)), n))}
%Y Cf. A058685, A058686.
%K sign
%O 0,8
%A _Michael Somos_, Aug 04 2011
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