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

Author

Ilya Gutkovskiy, Feb 18 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001615, A156303, A319111.

Programs

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)