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%I #51 Mar 12 2021 22:24:47
%S 1,-2,1,-2,3,-2,2,0,2,-2,1,-4,0,-2,3,-2,2,0,4,-2,2,0,0,-2,1,-4,2,-2,2,
%T -2,3,-2,0,-2,2,-2,2,0,2,-4,4,0,0,0,1,-2,4,0,2,-4,2,-2,1,-6,0,-2,2,0,
%U 0,-2,4,-2,0,-2,2,0,4,0,4,-2,1,-2,0,-2,4,0,0,-2
%N Expansion of f(-x, -x) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general 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="/A248886/b248886.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 f(-x)^2 * phi(-x^6) / phi(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.
%F Expansion of phi(-x) * phi(-x^6) / chi(-x^2) in powers of q where phi(), chi() are Ramanujan theta functions.
%F Expansion of q^(-1/12) * eta(q)^2 * eta(q^4) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [-2, 0, -2, -1, -2, -2, -2, -1, -2, 0, -2, -2, ...].
%F a(n) = (-1)^n * A123884(n). a(2*n) = A131961(n). a(2*n + 1) = -2 * A131963(n).
%e G.f. = 1 - 2*x + x^2 - 2*x^3 + 3*x^4 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + ...
%e G.f. = q - 2*q^13 + q^25 - 2*q^37 + 3*q^49 - 2*q^61 + 2*q^73 + 2*q^97 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^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^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2* eta(x^12 + A)), n))};
%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^4)*eta(q^6)^2/(eta(q^2)^2*eta(q^12))) \\ _Altug Alkan_, Jul 31 2018
%Y Cf. A123884, A131961, A131963.
%K sign
%O 0,2
%A _Michael Somos_, Oct 01 2015
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