A306483 Expansion of Product_{k>=1} 1/(1 - psi(k)*x^k), where psi() is the Dedekind psi function (A001615).
1, 1, 4, 8, 23, 41, 114, 200, 491, 909, 2036, 3710, 8235, 14743, 31058, 56538, 115435, 207401, 417876, 745578, 1470371, 2626489, 5086108, 9030162, 17347019, 30620651, 58060380, 102426652, 192288399, 337633825, 629845430, 1101958752, 2040109199, 3563507377, 6553539316, 11412799294
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
nmax = 35; CoefficientList[Series[Product[1/(1 - DirichletConvolve[i, MoebiusMu[i]^2, i, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] 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] 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}]
Formula
G.f.: exp(Sum_{k>=1} Sum_{j>=1} psi(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Feb 23 2019: (Start)
a(n) ~ c * 3^(n/2), where
c = 84.0923381459819921541124348082985... if n is even and
c = 82.6952907990079575265849718772977... if n is odd. (End)