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

0,3

Links

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

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)

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}]

Crossrefs

Cf. A001615, A156303, A319111.

Sequence in context: A075688 A272563 A272152 * A274521 A026596 A181688

Adjacent sequences: A306480 A306481 A306482 * A306484 A306485 A306486

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 18 2019

Status

approved