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A143066 Expansion of phi(x^3) / psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A143066 #15 Feb 16 2025 08:33:08
%S A143066 1,-1,1,0,1,-2,1,-1,2,-3,2,-1,4,-5,3,-3,6,-8,5,-4,9,-12,8,-7,14,-18,
%T A143066 13,-10,20,-26,18,-16,29,-37,27,-23,41,-52,38,-34,58,-72,54,-47,79,
%U A143066 -98,74,-67,109,-133,103,-92,146,-178,138,-127,196,-237,187,-170,260
%N A143066 Expansion of phi(x^3) / psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A143066 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A143066 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 10th equation.
%H A143066 G. C. Greubel, <a href="/A143066/b143066.txt">Table of n, a(n) for n = 0..1000</a>
%H A143066 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143066 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143066 Expansion of q^(-1/8) * eta(q) * eta(q^6)^5 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
%F A143066 Euler transform of period 12 sequence [ -1, 1, 1, 1, -1, -2, -1, 1, 1, 1, -1, 0, ...].
%F A143066 G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (2/3)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143068.
%F A143066 G.f.: (1 + 2 * x^3 + 2 * x^12 + 2 * x^27 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan]
%F A143066 G.f.: 1 - x * (1 - x) / (1 - x^4) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - x^9 * (1 - x) * (1 - x^3) * (1 - x^5) / ((1 - x^4) * (1 - x^8) * (1 - x^12)) + ... [Ramanujan]
%F A143066 -psi6 +2*psi3 -psi1
%F A143066 Expansion of psi(x^3)^2 / (psi(x) * psi(x^6)) in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Nov 08 2015
%F A143066 a(n) = A262929(4*n). a(3*n) = A262150(n). a(3*n + 1) = - A262152(n). a(3*n + 2) = A262157(n). - _Michael Somos_, Nov 08 2015
%e A143066 G.f. = 1 - x + x^2 + x^4 - 2*x^5 + x^6 - x^7 + 2*x^8 - 3*x^9 + 2*x^10 + ...
%e A143066 G.f. = 1/q - q^7 + q^15 + q^31 - 2*q^39 + q^47 - q^55 + 2*q^63 - 3*q^71 + ...
%t A143066 a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {-x^2}, x^2, x], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t A143066 a[ n_] := SeriesCoefficient[ 2 x^(1/8) EllipticTheta[ 3, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t A143066 a[ n_] := SeriesCoefficient[ x^(1/8)EllipticTheta[ 2, 0, x^(3/2)]^2 / (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3]), {x, 0, n}]; (* _Michael Somos_, Nov 08 2015 *)
%o A143066 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};
%Y A143066 Cf. A143068, A262150, A262152, A262157, A262929.
%K A143066 sign
%O A143066 0,6
%A A143066 _Michael Somos_, Jul 21 2008