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%I #12 Mar 12 2021 22:24:45
%S 1,-1,0,0,-1,0,0,0,0,0,2,0,2,0,0,0,-3,0,-4,0,0,0,4,0,5,0,0,0,-7,0,-8,
%T 0,0,0,12,0,14,0,0,0,-17,0,-20,0,0,0,24,0,28,0,0,0,-36,0,-40,0,0,0,52,
%U 0,56,0,0,0,-71,0,-80,0,0,0,96,0,109,0,0,0,-133
%N Expansion of q^(-1) * psi(-q) * phi(-q^9) / (psi(-q^3) * psi(q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A139216/b139216.txt">Table of n, a(n) for n = -1..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 eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 / (eta(q^2) * eta(q^3) * eta(q^12)^3 * eta(q^18)) in powers of q.
%F Euler transform of period 36 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, -2, 0, -1, 1, -1, 0, 0, -1, -1, -2, -1, -1, 0, 0, -1, 1, -1, 0, -2, -1, -1, -1, -1, -1, 0, 0, -1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139214.
%F a(n) = -(-1)^n * A139215(n). a(2*n) = 0 unless n=0.
%F a(3*n + 1) = 0. a(6*n + 3) = - A217786(n). - _Michael Somos_, Sep 07 2015
%e G.f. = 1/q - 1 - q^3 + 2*q^9 + 2*q^11 - 3*q^15 - 4*q^17 + 4*q^21 + 5*q^23 + ...
%t a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 4, 0, q^9] / (EllipticTheta[ 2, Pi/4, q^(3/2)] EllipticTheta[ 2, 0, q^3]), {q, 0, n}] // Simplify; (* _Michael Somos_, Sep 07 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^3 * eta(x^18 + A)), n))};
%Y Cf. A139214, A139215, A217786.
%K sign
%O -1,11
%A _Michael Somos_, Apr 11 2008
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