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%I #12 Mar 12 2021 22:24:44
%S 1,-1,1,-2,-1,0,1,1,2,-1,0,-1,0,1,-1,1,2,-2,1,-2,-3,0,0,1,2,0,1,-2,-2,
%T 2,0,2,3,-3,1,-3,-3,2,0,4,4,-2,0,-3,-3,2,-2,3,5,-3,1,-6,-6,2,0,5,6,-3,
%U 1,-4,-6,4,-2,6,7,-5,3,-8,-9,5,-1,7,9,-5,2,-8
%N Expansion of f(-x) * chi(x^2)^2 in powers of x where f(), 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="/A132966/b132966.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 q^(1/8) * eta(q^4)^4 * eta(q) / (eta(q^2)^2 * eta(q^8)^2) in powers of q.
%F Euler transform of period 8 sequence [ -1, 1, -1, -3, -1, 1, -1, -1, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 32^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132965.
%F G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k))^2 / (1 + x^(4*k))^2.
%e G.f. = 1 - x + x^2 - 2*x^3 - x^4 + x^6 + x^7 + 2*x^8 - x^9 - x^11 + x^13 + ...
%e G.f. = q^-1 - q^7 + q^15 - 2*q^23 - q^31 + q^47 + q^55 + 2*q^63 - q^71 - ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x^2, x^4]^2, {x, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^4 * eta(x + A) / (eta(x^2 + A)^2 * eta(x^8 + A)^2), n))};
%Y Cf. A132965.
%K sign
%O 0,4
%A _Michael Somos_, Aug 23 2007
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