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%I #9 Mar 12 2021 22:24:47
%S 1,-2,5,-10,18,-32,55,-90,145,-228,351,-532,795,-1170,1703,-2452,3494,
%T -4934,6910,-9598,13238,-18134,24680,-33390,44921,-60108,80029,
%U -106044,139875,-183706,240284,-313046,406319,-525490,677269,-870010,1114061,-1422210
%N Expansion of chi(x^2) / phi(x) in powers of x where phi(), 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="/A246712/b246712.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 phi(-x^4) / f(x)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%F Expansion of q^(1/12) * eta(q)^2 * eta(q^4)^4 / (eta(q^8) * eta(q^2)^6) in powers of q.
%F Euler transform of period 8 sequence [-2, 4, -2, 0, -2, 4, -2, 1, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085261.
%e G.f. = 1 - 2*x + 5*x^2 - 10*x^3 + 18*x^4 - 32*x^5 + 55*x^6 - 90*x^7 + ...
%e G.f. = 1/q - 2*q^11 + 5*q^23 - 10*q^35 + 18*q^47 - 32*q^59 + 55*q^71 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] / QPochhammer[ -x]^2, {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^4] / EllipticTheta[ 3, 0, x], {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)^4 / (eta(x^8 + A) * eta(x^2 + A)^6), n))};
%Y Cf. A085261.
%K sign
%O 0,2
%A _Michael Somos_, Sep 02 2014
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