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%I #12 Mar 12 2021 22:24:47
%S 1,-2,8,-16,32,-60,96,-160,256,-394,624,-944,1408,-2092,3008,-4320,
%T 6144,-8612,12072,-16720,22976,-31424,42528,-57312,76800,-102254,
%U 135728,-179104,235264,-307852,400704,-519808,671744,-864672,1109904,-1419456,1809568
%N Expansion of phi(q^2)^2 / (phi(q) * phi(q^4)) in powers of q where phi() is a Ramanujan 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="/A232358/b232358.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 Expansion of (eta(q) * eta(q^16) * eta(q^4)^7)^2 / (eta(q^2 ) * eta(q^8))^9 in powers of q.
%F Euler transform of period 16 sequence [-2, 7, -2, -7, -2, 7, -2, 2, -2, 7, -2, -7, -2, 7, -2, 0, ...].
%F G.f.: Product_{k>0} (1 + x^(4*k - 2))^5 / ((1 + x^(2*k - 1)) * (1 + x^(8*k - 4)))^2.
%F a(n) = (-1)^n * A212318(n). a(2*n) = A014969(n).
%F a(n) ~ (-1)^n * exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017
%e G.f. = 1 - 2*q + 8*q^2 - 16*q^3 + 32*q^4 - 60*q^5 + 96*q^6 - 160*q^7 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q^2, q^4]^5 / (QPochhammer[ -q, q^2] QPochhammer[ -q^4, q^8])^2, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A) * eta(x^4 + A)^7)^2 / (eta(x^2 + A) * eta(x^8 + A))^9, n))};
%Y Cf. A014969, A212318.
%K sign
%O 0,2
%A _Michael Somos_, Nov 23 2013
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