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%I #83 Sep 08 2022 08:46:08
%S 1,-8,25,-40,48,-80,121,-120,144,-200,192,-248,337,-280,336,-440,384,
%T -480,528,-480,673,-720,624,-720,816,-760,864,-1080,864,-1000,1321,
%U -1008,1200,-1360,1152,-1440,1536,-1400,1488,-1720,1536,-1760,2185,-1560,1872
%N Expansion of q^(-1/4) * eta(q)^8 * eta(q^4)^2 / eta(q^2)^5 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A244276/b244276.txt">Table of n, a(n) for n = 0..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 phi(-x)^4 * psi(x^2) = phi(-x)^3 * psi(-x)^2 = f(-x)^6 / phi(x) = psi(-x)^8 / psi(x^2)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%F Euler transform of period 4 sequence [ -8, -3, -8, -5, ...].
%F G.f.: Product_{k>0} (1 - x^k)^5 * (1 + x^(2*k))^2 / (1 + x^k)^3.
%F Convolution inverse of A134415.
%e G.f. = 1 - 8*x + 25*x^2 - 40*x^3 + 48*x^4 - 80*x^5 + 121*x^6 - 120*x^7 + ...
%e G.f. = q - 8*q^5 + 25*q^9 - 40*q^13 + 48*q^17 - 80*q^21 + 121*q^25 + ...
%t a[ n_] := SeriesCoefficient[QPochhammer[ x]^6/EllipticTheta[ 3, 0, x], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^4 EllipticTheta[ 2, 0, x]/(2 x^(1/4)), {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^3 EllipticTheta[ 2, Pi/4, x^(1/2)]^2/(2 x^(1/4)), {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^8 / (2 x^(1/4) EllipticTheta[ 2, 0, x]^3 ), {x, 0, n}];
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^8 * eta(x^ 4 + A)^2 / eta(x^2 + A)^5, n))};
%o (Magma) A := Basis( ModularForms( Gamma0(16), 5/2), 180); A[2] - 8*A[6];
%Y Cf. A134415.
%K sign
%O 0,2
%A _Michael Somos_, Sep 01 2014
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