A054192 Binomial transform of A000029.

1, 3, 8, 20, 49, 119, 289, 705, 1731, 4283, 10690, 26934, 68531, 176115, 457110, 1198128, 3170607, 8468277, 22818167, 61999531, 169778889, 468292663, 1300270333, 3632269293, 10202425207, 28798822159, 81652955889, 232429744843, 663969970203, 1902716831527

Offset: 0

Author

N. J. A. Sloane, Apr 29 2000

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Programs

Maple

with(numtheory):
b:= proc(n) option remember; ceil(add(
      phi(d)*2^(n/d)/(2*n), d=divisors(n))+
     `if`(n::odd, 2^((n-1)/2), 2^(n/2-1)+2^(n/2-2)))
    end:
a:= n-> add(b(n-j)*binomial(n, j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 17 2017

Mathematica

a29[n_] := If[n == 0, 1, DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n) + If[OddQ[n], 2^((n-1)/2), 2^(n/2-1) + 2^(n/2-2)]]; a[n_] := Sum[Binomial[n, k] * a29[k], {k, 0, n}]; Array[a, 28, 0] (* Jean-François Alcover, Jul 17 2017 *)

Formula

a(n) ~ 3^(n+1) / (4*n). - Vaclav Kotesovec, Nov 02 2023