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%I #10 Mar 12 2021 22:24:47
%S 1,-2,2,-4,3,-2,6,-4,4,-6,4,-4,7,-8,2,-8,8,-4,10,-4,4,-10,10,-8,9,-4,
%T 6,-12,8,-6,10,-12,4,-14,8,-4,16,-10,8,-8,9,-10,12,-12,8,-12,12,-4,20,
%U -10,6,-20,8,-6,10,-12,8,-20,18,-8,11,-12,12,-16,8,-6,20
%N Expansion of phi(-x) * psi(x^2)^2 in powers of x where phi(), psi() 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="/A246953/b246953.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 psi(x^2) * psi(-x)^2 = psi(-x)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
%F Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^4 / eta(q^2)^3 in powers of q.
%F Euler transform of period 4 sequence [ -2, 1, -2, -3, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128^(1/2) * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A246954.
%F G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4.
%F a(n) = (-1)^n * A045828(n). a(2*n) = A213625(n). a(2*n + 1) = - 2 * A213624(n).
%e G.f. = 1 - 2*x + 2*x^2 - 4*x^3 + 3*x^4 - 2*x^5 + 6*x^6 - 4*x^7 + 4*x^8 + ...
%e G.f. = q - 2*q^2 + 2*q^3 - 4*q^4 + 3*q^5 - 2*q^6 + 6*q^7 - 4*q^8 + 4*q^9 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x]^2/(4 x^(1/2)), {x, 0, n}];
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^4 / eta(x^2 + A)^3, n))};
%Y Cf. A045828, A213624, A213625, A246954.
%K sign
%O 0,2
%A _Michael Somos_, Sep 08 2014
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