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%I #12 Mar 12 2021 22:24:47
%S 1,1,0,-1,1,1,-2,-2,3,4,-4,-5,5,6,-8,-9,12,13,-14,-17,18,21,-26,-28,
%T 34,39,-42,-49,53,60,-70,-78,90,101,-110,-125,137,153,-174,-192,217,
%U 241,-264,-295,322,357,-400,-438,490,540,-588,-652,711,781,-866,-946
%N Expansion of f(-x^3, -x^5)^2 / (psi(-x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A245433/b245433.txt">Table of n, a(n) for n = 0..2500</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 Euler transform of period 8 sequence [1, -1, -1, 2, -1, -1, 1, 0, ...].
%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (u^2*v - v^2 - 2).
%F a(n) = A111374(2*n) = A245436(4*n - 1).
%e G.f. = 1 + x - x^3 + x^4 + x^5 - 2*x^6 - 2*x^7 + 3*x^8 + 4*x^9 + ...
%e G.f. = 1/q + q^3 - q^11 + q^15 + q^19 - 2*q^23 - 2*q^27 + 3*q^31 + ...
%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 1, -2, 1, 1, -1, 0}[[Mod[k, 8, 1]]], {k, n}], {x, 0, n}]; (* _Michael Somos_, Jun 27 2017 *)
%t f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x^3, -x^5]^2/(f[-x, -x^3]*f[x^2, x^6]), {x, 0, 60}], x]]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Aug 06 2018 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 1, 1, -2, 1, 1, -1][k%8 + 1]), n))};
%Y Cf. A111374, A245436.
%K sign
%O 0,7
%A _Michael Somos_, Jul 21 2014
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