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%I #8 Mar 12 2021 22:24:48
%S 0,1,-2,0,1,0,0,1,0,0,0,-2,0,1,-2,0,2,0,0,1,0,0,1,-2,0,0,0,0,1,-2,0,0,
%T 0,0,1,0,0,2,-2,0,1,-2,0,1,0,0,1,-2,0,1,0,0,1,0,0,0,-2,0,2,0,0,0,0,0,
%U 0,-2,0,1,-2,0,0,0,0,2,-2,0,1,0,0,3,0,0,1
%N Expansion of x * phi(-x) * psi(x^12) / chi(-x^3) in powers of x where phi(), psi(), chi() 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="/A260942/b260942.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 q^(-5/8) * eta(q)^2 * eta(q^6) * eta(q^24)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
%F Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, -2, ...].
%F a(3*n) = 0. a(3*n + 1) = A131963(n). a(3*n + 2) = -2 * A112609(n).
%e G.f. = x - 2*x^2 + x^4 + x^7 - 2*x^11 + x^13 - 2*x^14 + 2*x^16 + x^19 + ...
%e G.f. = q^13 - 2*q^21 + q^37 + q^61 - 2*q^93 + q^109 - 2*q^117 + 2*q^133 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6] QPochhammer[ -x^3, x^3] / (2 x^(1/2)), {x, 0, n}];
%t eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[ q^(-5/8)* eta[q]^2*eta[q^6]*eta[q^24]^2/(eta[q^2]*eta[q^3]*eta[q^12]), {q, 0, 50}], q] (* _G. C. Greubel_, Aug 01 2018 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^24 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
%Y Cf. A112609, A131963.
%K sign
%O 0,3
%A _Michael Somos_, Aug 04 2015
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