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%I #19 Mar 12 2021 22:24:45
%S 1,0,0,-1,1,0,0,-1,2,-1,0,-2,3,-1,0,-3,5,-2,1,-5,7,-3,1,-7,11,-5,2,
%T -11,15,-7,4,-15,22,-11,6,-22,30,-15,9,-30,42,-22,14,-42,56,-31,20,
%U -56,77,-43,29,-77,101,-58,41,-101,135,-80,57,-135,176,-106,78
%N Expansion of psi(-x^3) / f(-x^4) in powers of x where psi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 11th equation.
%H G. C. Greubel, <a href="/A143067/b143067.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 f(x, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%F Expansion of q^(-5/24) * eta(q^3) * eta(q^12) / (eta(q^4) * eta(q^6)) in powers of q.
%F Euler transform of period 12 sequence [ 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, ...].
%F G.f.: (1 + x + x^5 + x^8 + x^16 + x^21 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan]
%F G.f.: 1 - x^3 * (1 - x) / (1 - x^4) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - ... [Ramanujan]
%F a(2*n) = A262064(n). a(2*n + 3) = - A262090(n).
%F Convolution of A089801 and A106507. - _Michael Somos_, Jan 10 2017
%e G.f. = 1 - x^3 + x^4 - x^7 + 2*x^8 - x^9 - 2*x^11 + 3*x^12 - x^13 - 3*x^15 + ...
%e G.f. = q^5 - q^77 + q^101 - q^173 + 2*q^197 - q^221 - 2*q^269 + 3*q^293 + ...
%t a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {-x^2}, x^2, x^3], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ x^4], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t a[ n_] := SeriesCoefficient[ x^(-5/24) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]) / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* _Michael Somos_, Jan 10 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))};
%Y Cf. A089801, A106507, A262064, A262090.
%K sign
%O 0,9
%A _Michael Somos_, Jul 21 2008
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