%I #8 May 06 2019 07:09:44

%S 1,0,1,1,3,2,8,5,16,15,34,30,75,66,144,150,285,292,566,585,1062,1170,

%T 1988,2205,3729,4159,6755,7785,12214,14147,21957,25560,38709,45839,

%U 67884,80747,118332,141244,203614,245330,348396,420971,592439,717659,998248,1215439,1672544,2040210,2786687

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k-phi(k)), where phi() is the Euler totient function (A000010).

%C Euler transform of A051953.

%F G.f.: exp(Sum_{k>=1} (sigma_2(k) - sigma_2(k^2)/sigma_1(k^2)) * x^k/k).

%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} cototient(d^2) ) * x^k/k).

%F a(n) ~ exp(3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) + 1/4) * ((Pi^2 - 6)*Zeta(3))^(1/4) / (A^3 * 2^(1/12) * 3^(1/2) * Pi^(5/6) * n^(3/4)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, May 06 2019

%t nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(k - EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 48; CoefficientList[Series[Exp[Sum[(DivisorSigma[2, k] - DivisorSigma[2, k^2]/DivisorSigma[1, k^2]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 - EulerPhi[d^2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

%Y Cf. A000010, A001157, A051953, A053650, A057660, A061255, A065764, A065827.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Apr 22 2019