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%I #12 Sep 08 2022 08:46:09
%S 1,-1,0,4,-5,0,4,-8,0,2,-4,0,12,-4,0,16,-13,0,0,-12,0,8,-4,0,20,-5,0,
%T 4,-16,0,8,-24,0,8,-8,0,10,-4,0,32,-20,0,8,-12,0,0,-8,0,28,-9,0,24,
%U -20,0,4,-32,0,8,-4,0,32,-12,0,16,-29,0,16,-12,0,16,-8
%N Expansion of psi(-q) * phi(q^3)^2 * chi(q^3) in powers of q where phi(), psi(), chi() 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="/A246927/b246927.txt">Table of n, a(n) for n = 0..2500</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)^12 / (eta(q^2) * eta(q^3)^5 * eta(q^12)^5) in powers of q.
%F Euler transform of period 12 sequence [-1, 0, 4, -1, -1, -7, -1, -1, 4, 0, -1, -3, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 246^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246926.
%F a(3*n) = A034933(n). a(3*n + 1) = - A246926(n). a(3*n + 2) = 0.
%e G.f. = 1 - q + 4*q^3 - 5*q^4 + 4*q^6 - 8*q^7 + 2*q^9 - 4*q^10 + 12*q^12 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 3, 0, q^3]^2 / (2^(1/2) q^(1/8)), {q, 0, n}];
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^12 / (eta(x^2 + A) * eta(x^3 + A)^5 * eta(x^12 + A)^5), n))};
%o (Magma) A := Basis( ModularForms( Gamma0(36), 3/2), 70); A[1] - A[2] + 4*A[4] - 5*A[5] + 4*A[6];
%Y Cf. A034933, A246926.
%K sign
%O 0,4
%A _Michael Somos_, Sep 07 2014
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