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%I #27 Mar 12 2021 22:24:48
%S 1,-2,2,-5,9,-12,16,-23,36,-47,60,-84,115,-149,188,-245,321,-406,505,
%T -641,813,-1007,1237,-1533,1901,-2321,2816,-3437,4191,-5055,6068,
%U -7307,8792,-10501,12490,-14886,17720,-20975,24755,-29236,34492,-40522,47486,-55666
%N Expansion of phi(-x^3) * f(-x, -x^5) / psi(x) in powers of x where phi(), psi(), f(, ) 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="/A262614/b262614.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 f(-x^3)^3 / (f(x, x^2) * psi(x)) in powers of x where psi(), f(, ) are Ramanujan theta functions.
%F Expansion of q^(-5/24) * eta(q)^2 * eta(q^3) * eta(q^6) / eta(q^2)^3 in powers of q.
%F Euler transform of period 6 sequence [ -2, 1, -3, 1, -2, -1, ...].
%F a(n) = A053269(3*n + 1).
%F a(n) ~ (-1)^n * exp(sqrt(n/2)*Pi) / (6*sqrt(n)). - _Vaclav Kotesovec_, Apr 17 2016
%e G.f. = 1 - 2*x + 2*x^2 - 5*x^3 + 9*x^4 - 12*x^5 + 16*x^6 - 23*x^7 +
%e G.f. = q^5 - 2*q^29 + 2*q^53 - 5*q^77 + 9*q^101 - 12*q^125 + 16*q^149 - 23*q^173 + ...
%t a[ n_] := SeriesCoefficient[ 2 x^(1/8) QPochhammer[ x^3]^3 QPochhammer[ x, x^2] / (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)]), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A)^3, n))};
%Y Cf. A053269.
%K sign
%O 0,2
%A _Michael Somos_, Apr 17 2016
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