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%I #9 Mar 12 2021 22:24:46
%S 1,-2,2,-2,0,2,-4,6,-4,0,6,-12,14,-10,0,14,-26,30,-22,0,28,-52,60,-42,
%T 0,54,-100,112,-78,0,100,-180,202,-140,0,174,-314,350,-240,0,296,-532,
%U 588,-402,0,492,-876,966,-658,0,794,-1412,1550,-1050,0,1260,-2232
%N Expansion of f(-x, -x^4) / f(x, x^4) in powers of x where f(,) is Ramanujan's two-variable 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="/A215594/b215594.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 Euler transform of period 10 sequence [ -2, 1, 0, -2, 0, -2, 0, 1, -2, 0, ...].
%F a(5*n + 4) = 0.
%e 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^5 - 4*x^6 + 6*x^7 - 4*x^8 + 6*x^10 - 12*x^11 + ...
%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A215594[n_] := SeriesCoefficient[f[-x, -x^4]/f[x, x^4], {x, 0, n}]; Table[A215594[n], {n,0,50}] (* _G. C. Greubel_, Jun 18 2017 *)
%o (PARI) {a(n) = local(A, s); if( n<0, 0, A = x * O(x^n); s = sqrtint( 40*n + 9); polcoeff( sum( k=(-s + 6)\10, (s - 3)\10, (-1)^k * x^((5*k + 3)*k/2), A) / sum( k=(-s + 6)\10, (s - 3)\10, x^((5*k + 3)*k/2), A), n))}
%K sign
%O 0,2
%A _Michael Somos_, Aug 16 2012
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