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%I #6 Sep 08 2022 08:46:23
%S 1,2,-4,-8,6,8,-8,0,12,10,-8,-24,8,8,-16,0,6,16,-12,-24,24,16,-8,0,24,
%T 10,-24,-32,0,24,-16,0,12,16,-16,-48,30,8,-24,0,24,32,-16,-24,24,24,
%U -16,0,8,18,-28,-48,24,24,-32,0,48,16,-8,-72,0,24,-32,0,6,32
%N Expansion of phi(-q) * phi(q)^2 in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%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^2)^9 / (eta(q)^2 * eta(q^4)^4) in powers of q.
%F Expansion of phi(q) * phi(-q^2)^2 = phi(-q^2)^4 / phi(-q) in powers of q.
%F Euler transform of period 4 sequence [2, -7, 2, -3, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(11/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045834.
%F G.f. Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5 / (1 + x^(2*k))^4.
%F a(n) = (-1)^n * A212885(n) = A083703(2*n) = A080965(2*n).
%F a(4*n) = a(n) * -A132429(n + 2) where A132429 is a period 4 sequence.
%F a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = -4 * A045828(n).
%F a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = -4 * A213625(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 4) = A005887(n). a(8*n + 5) = 2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.
%e G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 6*x^4 + 8*x^5 - 8*x^6 + 12*x^8 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};
%o (Magma) A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];
%Y Cf. A004015, A004024, A005887, A008443, A045834, A083703, A080965, A132429, A212885, A213624.
%K sign
%O 0,2
%A _Michael Somos_, Sep 09 2018
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