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%I #15 Mar 12 2021 22:24:47
%S 1,-2,1,-2,3,-2,4,-4,2,-2,5,-4,2,-6,3,-6,7,-2,5,-4,5,-6,6,-2,5,-10,3,
%T -6,10,-4,6,-8,3,-8,7,-6,7,-6,4,-6,11,-6,9,-10,3,-6,14,-4,8,-10,8,-10,
%U 5,-6,4,-16,7,-4,10,-4,13,-14,7,-8,8,-6,10,-12,7,-12
%N Expansion of psi(-x)^2 * psi(x^4) in powers of x where psi() 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="/A246833/b246833.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^(-3/4) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^2 in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246815.
%F a(n) = (-1)^n * A213624(n). a(2*n) = A246832(n). a(2*n + 1) = -2 * A033763(n).
%e G.f. = 1 - 2*x + x^2 - 2*x^3 + 3*x^4 - 2*x^5 + 4*x^6 - 4*x^7 + 2*x^8 - 2*x^9 + ...
%e G.f. = q^3 - 2*q^7 + q^11 - 2*q^15 + 3*q^19 - 2*q^23 + 4*q^27 - 4*q^31 + 2*q^35 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^2 EllipticTheta[ 2, 0, x^2] / (4 x^(3/4)), {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^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))};
%Y Cf. A033763, A213624, A246815, A246832.
%K sign
%O 0,2
%A _Michael Somos_, Sep 04 2014
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