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%I #26 Mar 12 2021 22:24:45
%S 1,6,28,104,342,1016,2808,7296,18044,42750,97656,215992,464360,973176,
%T 1993328,3998592,7870038,15221232,28968084,54311736,100421688,
%U 183281904,330468216,589084288,1038850488,1813500030,3135518440,5372110496,9124793472,15371832424
%N Expansion of eta(q^4)^2 / (eta(q^2) * eta(q)^6) 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="/A134416/b134416.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 Euler transform of period 4 sequence [ 6, 7, 6, 5, ...].
%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k)^5. [corrected by _Vaclav Kotesovec_, Sep 07 2015]
%F a(n) ~ exp(2*Pi*sqrt(n))/(32*n^2). - _Vaclav Kotesovec_, Sep 07 2015
%F -2 * a(n) = A134414(4*n).
%F Expansion of psi(q^2) / f(-q)^6 = phi(q)^3 / phi(-q^2)^8 = 1 / (phi(-q)^3 * phi(-q^2)^2) = 1 / (phi(q) * phi(-q)^4) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Oct 16 2015
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-13/2) (t/i)^(-5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134414. - _Michael Somos_, Oct 16 2015
%F Convolution inverse is A245643. - _Michael Somos_, Oct 16 2015
%e G.f. = 1 + 6*q + 28*q^2 + 104*q^3 + 342*q^4 + 1016*q^5 + 2808*q^6 + 7296*q^7 + ...
%t nmax = 40; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k)^5, {k, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t a[ n_] := SeriesCoefficient[ 1 / (EllipticTheta[ 4, 0 , q]^3 EllipticTheta[ 4, 0, q^2]^2), {q, 0, n}]; (* _Michael Somos_, Oct 16 2015 *)
%t a[ n_] := SeriesCoefficient[ 1 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* _Michael Somos_, Oct 16 2015 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 4, 0, q^2]^8, {q, 0, n}]; (* _Michael Somos_, Oct 16 2015 *)
%t QP = QPochhammer; s = QP[q^4]^2/(QP[q^2]*QP[q]^6) + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x^2 + A) * eta(x + A)^6), n))};
%o (PARI) q='q+O('q^99); Vec(eta(q^4)^2/(eta(q^2)*eta(q)^6)) \\ _Altug Alkan_, Apr 16 2018
%Y Cf. A134414, A245643.
%K nonn
%O 0,2
%A _Michael Somos_, Oct 26 2007
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