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%I A138559 #19 Mar 12 2021 22:24:45
%S A138559 1,1,-2,-1,1,-1,-1,-1,2,2,-2,0,1,1,-1,0,3,1,-3,-2,3,0,-2,-1,3,2,-4,-2,
%T A138559 2,1,-4,-2,5,3,-6,-1,5,1,-5,-3,6,3,-6,-3,7,2,-6,-2,9,5,-10,-5,9,3,-9,
%U A138559 -4,11,6,-12,-4,11,5,-12,-5,14,6,-16,-7,15,5,-16,-7,19,9,-20,-8,19,7,-20,-10,24,11,-25,-11,24,9,-26,-11,29,13,-31,-13
%N A138559 Expansion of phi(x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A138559 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138559 Vaclav Kotesovec, <a href="/A138559/b138559.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H A138559 O-Yeat Chan, <a href="https://doi.org/10.4064/aa120-2-1">Some asymptotics for cranks</a>, Acta Arithmetica 120 (2005), p. 107-143.
%H A138559 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138559 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138559 Expansion of phi(x) * chi(-x) = phi(-x^2) * chi(x) = psi(x) * chi(-x^2)^2 = f(-x) * chi(x)^2 = f(x) * chi(-x^2) = f(x)^2 / psi(x) = phi(-x^2)^2 / f(-x) if powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A138559 Expansion of q^(1/24) * eta(q^2)^4 / (eta(q^4)^2 * eta(q)) in powers of q.
%F A138559 Euler transform of period 4 sequence [ 1, -3, 1, -1, ...].
%F A138559 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A096920.
%F A138559 G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2.
%F A138559 a(n) ~ c1(n) * exp(Pi*sqrt(2*(n - 1/24)/3)/4) / (2*sqrt(2*(n - 1/24))), where c1(n) = (-1)^(n/2) * 1.847759... if n is even and c1(n) = -(-1)^((n+1)/2) * 0.765366... if n is odd [Chan, 2005, formula 2.2 and 2.7, where a1(n) = a(n) if n is even and a1(n) = -a(n) if n is odd]. - _Vaclav Kotesovec_, May 08 2020
%F A138559 In closed form, abs(c1(n)) = sqrt(2 + (-1)^n*sqrt(2)). - _Vaclav Kotesovec_, May 08 2020
%e A138559 G.f. = 1 + x - 2*x^2 - x^3 + x^4 - x^5  x^6 - x^7 + 2*x^8 + 2*x^9 - 2*x^10 + ...
%e A138559 G.f. = 1/q + q^23 - 2*q^47 - q^71 + q^95 - q^119 - q^143 - q^167 + 2*q^191 + ...
%t A138559 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, Aug 31 2014 *)
%t A138559 CoefficientList[Series[QPochhammer[-x] * QPochhammer[x^2] / QPochhammer[x^4], {x, 0, 100}], x] (* _Vaclav Kotesovec_, May 08 2020 *)
%o A138559 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)^2), n))};
%Y A138559 Cf. A096920.
%K A138559 sign
%O A138559 0,3
%A A138559 _Michael Somos_, Mar 24 2008

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