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Expansion of psi(x^2)^2 * phi(-x^2)^6 + 8 * x * psi(x^2)^6 * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
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%I #32 Mar 12 2021 22:24:47

%S 1,8,-10,16,37,-40,-50,-80,-30,40,128,48,-25,80,-34,320,-320,-160,310,

%T -400,410,152,-370,-416,-87,-240,-410,400,320,-200,30,592,500,776,384,

%U 400,-630,-200,-640,-1120,-359,552,300,-272,-326,-800,2560,-400,-110

%N Expansion of psi(x^2)^2 * phi(-x^2)^6 + 8 * x * psi(x^2)^6 * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan 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="/A228072/b228072.txt">Table of n, a(n) for n = 0..1000</a>

%H Hossein Movasati, Younes Nikdelan, <a href="https://arxiv.org/abs/1803.01414">Product formulas for weight two newforms</a>, arXiv:1803.01414 [math.NT], 2018.

%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^(-1/2) * ((eta(q^2)^5 / eta(q^4))^2 + 8 * (eta(q^4)^5 / eta(q^2))^2) in powers of q.

%F Expansion of q^(-1/2) * (eta(q^2)^12 + 8 * eta(q^4)^12) / ( eta(q^2) * eta(q^4) )^2 in powers of q.

%F a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^2 (t / i)^4 f(t) where q = exp(2 Pi i t).

%F a(2*n) = A227695(n). a(2*n + 1) = 8 * A227317(n).

%F If F(x) is the g.f. for A002171, then A(x) * F(x^2) = B(x) the g.f. for A227239. - _Michael Somos_, Jan 08 2015

%e G.f. = 1 + 8*x - 10*x^2 + 16*x^3 + 37*x^4 - 40*x^5 - 50*x^6 - 80*x^7 - 30*x^8 + ...

%e G.f. = q + 8*q^3 - 10*q^5 + 16*q^7 + 37*q^9 - 40*q^11 - 50*q^13 - 80*q^15 - 30*q^17 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^12 + 8 x QPochhammer[ x^4]^12) / (QPochhammer[ x^2] QPochhammer[ x^4])^2, {x, 0, n}];

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n);polcoeff( (eta(x^2 + A)^5 / eta(x^4 + A))^2 + 8 * x * (eta(x^4 + A)^5 / eta(x^2 + A))^2, n))};

%Y Cf. A002171, A227239, A227317, A227695.

%K sign

%O 0,2

%A _Michael Somos_, Sep 02 2013