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A306483 Expansion of Product_{k>=1} 1/(1 - psi(k)*x^k), where psi() is the Dedekind psi function (A001615).

%I A306483 #8 Feb 23 2019 09:02:16
%S A306483 1,1,4,8,23,41,114,200,491,909,2036,3710,8235,14743,31058,56538,
%T A306483 115435,207401,417876,745578,1470371,2626489,5086108,9030162,17347019,
%U A306483 30620651,58060380,102426652,192288399,337633825,629845430,1101958752,2040109199,3563507377,6553539316,11412799294
%N A306483 Expansion of Product_{k>=1} 1/(1 - psi(k)*x^k), where psi() is the Dedekind psi function (A001615).
%H A306483 Vaclav Kotesovec, <a href="/A306483/b306483.txt">Table of n, a(n) for n = 0..1000</a>
%F A306483 G.f.: exp(Sum_{k>=1} Sum_{j>=1} psi(j)^k*x^(j*k)/k).
%F A306483 From _Vaclav Kotesovec_, Feb 23 2019: (Start)
%F A306483 a(n) ~ c * 3^(n/2), where
%F A306483 c = 84.0923381459819921541124348082985... if n is even and
%F A306483 c = 82.6952907990079575265849718772977... if n is odd. (End)
%t A306483 nmax = 35; CoefficientList[Series[Product[1/(1 - DirichletConvolve[i, MoebiusMu[i]^2, i, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A306483 nmax = 35; CoefficientList[Series[Exp[Sum[Sum[DirichletConvolve[i, MoebiusMu[i]^2, i, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t A306483 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DirichletConvolve[i, MoebiusMu[i]^2, i, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]
%Y A306483 Cf. A001615, A156303, A319111.
%K A306483 nonn
%O A306483 0,3
%A A306483 _Ilya Gutkovskiy_, Feb 18 2019