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%I #8 Mar 12 2021 22:24:48
%S 1,-1,-1,1,1,-2,-1,2,1,-1,-2,2,1,0,-2,2,1,0,-1,0,2,-2,-2,0,1,-3,0,1,2,
%T -2,-2,2,1,-2,0,4,1,0,0,0,2,0,-2,0,2,-2,0,0,1,-3,-3,0,0,-2,-1,4,2,0,
%U -2,2,2,0,-2,2,1,0,-2,0,0,0,-4,0,1,-2,0,3,0,-4
%N Expansion of f(-x) * f(x^4, x^8)^2 / f(-x^3, -x^9) 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 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 phi(-x^12)^2 * psi(-x^2)^2 / (psi(x) * psi(-x^3)) in powers of x where phi(), psi() are Ramanujan theta functions.
%F Expansion of eta(q) * eta(q^6) * eta(q^8)^2 * eta(q^12)^3 / (eta(q^3) * eta(q^4)^2 * eta(q^24)^2) in powers of q.
%F Euler transform of period 24 sequence [ -1, -1, 0, 1, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, 1, 0, -1, -1, -2, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 384^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261119.
%F a(n) = (-1)^(n + floor(n/2)) * A000377(n) = (-1)^floor(n/2) * A190611(n).
%F a(2*n) = A190611(n). a(2*n + 1) = - A190615(n). a(4*n) = A000377(n). a(4*n + 1) = - A261118(n). a(4*n + 2) = - A129402(n). a(4*n + 3) - A261119(n).
%e G.f. = 1 - x - x^2 + x^3 + x^4 - 2*x^5 - x^6 + 2*x^7 + x^8 - x^9 + ...
%t a[ n_] := SeriesCoefficient[ 2^(1/2) EllipticTheta[ 4, 0, x^12]^2 EllipticTheta[ 2, Pi/4, x]^2 / (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)]), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^8 + A)^2 * eta(x^12 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^24 + A)^2), n))};
%Y Cf. A000377, A129402, A190611, A190615, A261118, A261119.
%K sign
%O 0,6
%A _Michael Somos_, Aug 09 2015
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