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Expansion of psi(x) * psi(x^9) * f(-x^3) / psi(x^3)^2 in powers of x where psi(), and f() are Ramanujan theta functions.
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%I #18 Mar 12 2021 22:24:48

%S 1,1,0,-2,-3,0,2,4,0,-5,-5,0,9,8,0,-12,-14,0,16,20,0,-23,-25,0,36,37,

%T 0,-47,-54,0,60,71,0,-84,-91,0,115,121,0,-149,-164,0,188,210,0,-245,

%U -264,0,321,343,0,-406,-443,0,505,554,0,-641,-687,0,813,863,0

%N Expansion of psi(x) * psi(x^9) * f(-x^3) / psi(x^3)^2 in powers of x where psi(), and 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%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13, 10th equation.

%H G. C. Greubel, <a href="/A267852/b267852.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) * f(-x^6) * f(-x^3, -x^15) / f(-x, -x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%F Expansion of q^(-5/8) * eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^2 / (eta(q) * eta(q^6)^4 * eta(q^9)) in powers of q.

%F Euler transform of period 18 sequence [ 1, -1, -2, -1, 1, 0, 1, -1, -1, -1, 1, 0, 1, -1, -2, -1, 1, -1, ...].

%F a(3*n) = A262614(n). a(3*n + 1) = A263041(n). a(3*n + 2) = 0.

%e G.f. = 1 + x - 2*x^3 - 3*x^4 + 2*x^6 + 4*x^7 - 5*x^9 - 5*x^10 + 9*x^12 + ...

%e G.f. = q^5 + q^13 - 2*q^29 - 3*q^37 + 2*q^53 + 4*q^61 - 5*q^77 - 5*q^85 + ...

%t a[ n_] := SeriesCoefficient[ x^(-1/2) QPochhammer[ x^3] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(3/2)]^2, {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^2 / (eta(x + A) * eta(x^6 + A)^4 * eta(x^9 + A)), n))};

%Y Cf. A053269, A262614, A263041.

%K sign

%O 0,4

%A _Michael Somos_, Apr 17 2016