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%I #12 May 28 2019 19:29:21
%S 1,1,3,15,93,765,6615,73395,855225,11348505,163593675,2633729175,
%T 44537325525,829112008725,16299062754975,340762189642875,
%U 7597436750528625,178862527106888625,4426363064514265875,115222810432347993375,3139125774622690978125
%N Expansion of e.g.f. (1/(1 - x)) * Product_{k>=2} 1/(1 - x^k)^(phi(k)/2), where phi() is the Euler totient function (A000010).
%H Vaclav Kotesovec, <a href="/A308457/b308457.txt">Table of n, a(n) for n = 0..435</a>
%F E.g.f.: exp(Sum_{k>=1} A057661(k)*x^k/k).
%F E.g.f.: exp(Sum_{k>=1} A051193(k)*x^k/k^2).
%F E.g.f.: d/dx ( exp(arctanh(x)) ) * Product_{k>=3} 1/(1 - x^k)^A023022(k).
%F a(n) ~ A * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi)^(2/3) - n - 1/12) * n^(n + 1/36) / (2^(1/9) * 3^(19/36) * (Pi*Zeta(3))^(1/36)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, May 28 2019
%F E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A023896(k)/k). - _Ilya Gutkovskiy_, May 28 2019
%t nmax = 20; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^(EulerPhi[k]/2), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 20; CoefficientList[Series[Exp[Sum[Sum[LCM[k, j], {j, 1, k}] x^k/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Total[Numerator[Range[k]/k]] k! Binomial[n - 1, k - 1] a[n - k]/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]
%Y Cf. A000010, A023022, A023896, A051193, A057661, A061255.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 27 2019
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