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%I #26 Apr 03 2019 02:58:24
%S 1,2,6,18,52,146,402,1090,2916,7708,20160,52236,134222,342304,867024,
%T 2182384,5461696,13595918,33677550,83036878,203859820,498470998,
%U 1214230586,2947204870,7129403128,17191258642,41328057106,99067295658,236822823336,564650823162,1342921372126
%N Expansion of Product_{k>=1} ((1 - x)^k + x^k)/((1 - x)^k - x^k).
%C First differences of the binomial transform of A015128.
%C Convolution of A129519 and A218482.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: 1/theta_4(x/(1 - x)), where theta_4() is the Jacobi theta function.
%F G.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/(k*(1 - x)^k)).
%F a(n) ~ 2^(n-3) * exp(Pi*sqrt(n/2) + Pi^2/16) / n. - _Vaclav Kotesovec_, Oct 15 2018
%p a:=series(mul(((1-x)^k+x^k)/((1-x)^k-x^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 30; CoefficientList[Series[Product[((1 - x)^k + x^k)/((1 - x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 30; CoefficientList[Series[1/EllipticTheta[4, 0, x/(1 - x)], {x, 0, nmax}], x]
%t nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
%Y Cf. A000203, A015128, A129519, A218482, A266497.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 15 2018
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