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%I A131126 #29 Mar 12 2021 04:04:41
%S A131126 1,4,16,48,128,312,704,1504,3072,6036,11488,21264,38400,67864,117632,
%T A131126 200352,335872,554952,904784,1457136,2320128,3655296,5702208,8813472,
%U A131126 13504512,20523996,30952544,46340832,68901888,101777112,149403264,218016640,316342272
%N A131126 Expansion of (phi(q^2) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.
%C A131126 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A131126 G. C. Greubel, <a href="/A131126/b131126.txt">Table of n, a(n) for n = 0..1000</a>
%H A131126 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A131126 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A131126 Expansion of ((phi(q) / phi(-q))^2 + 1) / 2 in powers of q where phi() is a Ramanujan theta function.
%F A131126 Expansion of (eta(q^4)^5 / (eta(q)^2 * eta(q^2) * eta(q^8)^2))^2 in powers of q.
%F A131126 Euler transform of period 8 sequence [ 4, 6, 4, -4, 4, 6, 4, 0, ...].
%F A131126 a(n) = 4 * A107035(n) unless n=0. 2 * a(n) = A014969(n) unless n=0.
%F A131126 a(n) ~ exp(sqrt(2*n)*Pi) / (16 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%F A131126 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/2 + (1/8)*sqrt(8 + 6*sqrt(2)). - _Simon Plouffe_, Mar 04 2021
%e A131126 G.f. = 1 + 4*q + 16*q^2 + 48*q^3 + 128*q^4 + 312*q^5 + 704*q^6 + 1504*q^7 + ...
%t A131126 nmax = 50; CoefficientList[Series[Product[((1 - x^(4*k))^5 / ((1 - x^k)^2 * (1 - x^(2*k)) * (1 - x^(8*k))^2))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%t A131126 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%o A131126 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5)^-2, n))};
%Y A131126 Cf. A014969, A107035.
%K A131126 nonn
%O A131126 0,2
%A A131126 _Michael Somos_, Jun 15 2007

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