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%I #14 Mar 12 2021 22:24:45
%S 1,1,1,0,0,1,1,2,1,1,1,2,3,3,3,2,2,3,5,6,5,4,4,6,9,10,9,8,8,11,14,16,
%T 15,13,14,18,24,26,25,22,23,29,36,40,38,36,38,46,56,61,60,56,59,70,84,
%U 92,90,86,90,106,125,135,134,130,136,157,181,196,195
%N Expansion of chi(-x^3)^2 / (chi(-x) * chi(-x^9)) in power of x 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="/A134132/b134132.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 q^(-1/6) * eta(q^2) * eta(q^3)^2 * eta(q^18) / (eta(q) * eta(q^6)^2 * eta(q^9)) in powers of q.
%F Euler transform of period 18 sequence [ 1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, 1, 0, ...].
%F Given g.f. A(x) then B(q) = A(q^6) * q satisfies 0 = f(B(q), B(q^2), B(q^4) ) where f(u, v, w) = (u^2 + v) * w^2 - (u^2 - v) * v.
%F Given g.f. A(x) then B(q) = A(q^3)^2 * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) * (1 + u * v) - (2 * u * v)^2.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (648 t)) = 1 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134131.
%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2.
%F -a(n) = A112178(3*n + 1). Convolution inverse of A134131.
%F a(n) ~ exp(2*Pi*sqrt(n/3)/3) / (2 * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e G.f. = 1 + x + x^2 + x^5 + x^6 + 2*x^7 + x^8 + x^9 + x^10 + 2*x^11 + 3*x^12 + ...
%e G.f. = q + q^7 + q^13 + q^31 + q^37 + 2*q^43 + q^49 + q^55 + q^61 + 2*q^67 + ...
%t nmax = 80; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3, x^6]^2 QPochhammer[ -x^9, x^9] , {x, 0, n}]; (* _Michael Somos_, Oct 27 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)), n))};
%Y Cf. A112178, A134131.
%K nonn
%O 0,8
%A _Michael Somos_, Oct 10 2007, Oct 21 2007
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