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%I #13 Mar 12 2021 22:24:47
%S 1,0,1,-2,2,-2,3,-4,5,-6,7,-10,13,-14,17,-22,26,-30,36,-44,52,-60,70,
%T -84,99,-112,131,-156,179,-204,236,-274,315,-358,409,-472,539,-608,
%U 692,-792,897,-1010,1144,-1298,1464,-1644,1849,-2088,2347,-2622,2940,-3304
%N Expansion of phi(-x^3) / f(-x^2) in powers of x where phi(), f() 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="/A256636/b256636.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, x^2) / psi(x) in powers of x where psi(), f() are Ramanujan theta functions.
%F Expansion of q^(1/12) * eta(q^3)^2 / (eta(q^2) * eta(q^6)) in powers of q.
%F Euler transform of period 6 sequence [ 0, 1, -2, 1, 0, 0, ...].
%F G.f.: Product_{k>0} (1 + x^k + x^(2*k)) / ((1 + x^k) * (1 + x^(3*k))). [corrected by _Vaclav Kotesovec_, Jul 11 2016]
%F a(n) = (-1)^n * A258327(n). - _Michael Somos_, Jun 06 2015
%e G.f. = 1 + x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 3*x^6 - 4*x^7 + 5*x^8 - 6*x^9 + ...
%e G.f. = 1/q + q^23 - 2*q^35 + 2*q^47 - 2*q^59 + 3*q^71 - 4*q^83 + 5*q^95 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / QPochhammer[ x^2], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 / (QPochhammer[ x^2] QPochhammer[ x^6]), {x, 0, n}]; (* _Michael Somos_, Jun 06 2015 *)
%t a[ n_] := SeriesCoefficient[ 2 x^(1/8) EllipticTheta[ 4, 0, x^3] / (EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ x, x^2]), {x, 0, n}]; (* _Michael Somos_, Jun 06 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)), n))};
%Y Cf. A258327.
%K sign
%O 0,4
%A _Michael Somos_, Apr 06 2015
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