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%I #31 Mar 12 2021 22:24:48
%S 1,-2,2,-4,6,-8,12,-16,22,-28,36,-48,60,-76,96,-120,150,-184,228,-280,
%T 340,-416,504,-608,732,-878,1052,-1252,1488,-1768,2088,-2464,2902,
%U -3408,3996,-4672,5460,-6364,7400,-8600,9972,-11544,13344,-15400,17752,-20424
%N Expansion of chi(-q) * chi(q^9) / (chi(q) * chi(-q^9)) in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Vaclav Kotesovec, <a href="/A260215/b260215.txt">Table of n, a(n) for n = 0..2000</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 psi(-q) * psi(q^9) / (psi(q) * psi(-q^9)) in powers of q where psi() is a Ramanujan theta function.
%F Expansion of eta(q)^2 * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9)^2 * eta(q^36)) in powers of q.
%F Euler transform of period 36 sequence [ -2, 1, -2, 0, -2, 1, -2, 0, 0, 1, -2, 0, -2, 1, -2, 0, -2, 0, -2, 0, -2, 1, -2, 0, -2, 1, 0, 0, -2, 1, -2, 0, -2, 1, -2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A128143.
%F a(n) = (-1)^n * A261156(n). Convolution inverse of A261156
%F a(2*n + 1) = -2 * A261203(n) = -2 * A261154(2*n + 1). 2 * a(2*n) = A261154(2*n) unless n=0.
%F a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = 2 * A260057(n). - _Michael Somos_, Nov 08 2015
%F a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e G.f. = 1 - 2*x + 2*x^2 - 4*x^3 + 6*x^4 - 8*x^5 + 12*x^6 - 16*x^7 + 22*x^8 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q, -q] QPochhammer[ -q^9, q^18] QPochhammer[ -q^9, q^9], {q, 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) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A)), n))};
%Y Cf. A128143, A260057, A261154, A261156, A261203, A261230, A261325.
%K sign
%O 0,2
%A _Michael Somos_, Aug 13 2015
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