A095975 -a(n) is inverse EULER transform of -A000041(n).

1, 2, 5, 11, 27, 60, 147, 344, 839, 2031, 5017, 12379, 30921, 77407, 195121, 493451, 1253613, 3194303, 8166757, 20933754, 53798919, 138566312, 357647565, 924834079, 2395702801, 6215748612, 16150985071, 42024182520, 109485000777, 285578913962, 745728542725

Offset: 1

Author

Vladeta Jovovic, Jul 20 2004

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A055887, A246828.

Programs

Maple

with(numtheory): b:= proc(n) option remember; `if`(n=0,1, add(add(d, d=divisors(j)) *b(n-j), j=1..n)/n) end: c:= proc(n) option remember; local j; add(c(j) *b(n-j), j=1..n-1)-n*b(n) end: a:= -proc(n) option remember; local d; `if`(n=0,1, add(mobius(n/d)*c(d), d=divisors(n))/n) end: seq(a(n), n=1..40); # Alois P. Heinz, Sep 09 2008
# The function EulerInvTransform is defined in A358451.
a := -EulerInvTransform(n -> -combinat:-numbpart(n)):
seq(a(n), n = 1..31); # Peter Luschny, Nov 21 2022

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d, {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; c[n_] := c[n] = Sum[c[j]*b[n-j], {j, 1, n-1}] - n*b[n]; a[n_] := -If[n == 0, 1, Sum[MoebiusMu[n/d]*c[d], {d, Divisors[n]}]/n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Formula

Moebius transform of A055890(n).

a(n) ~ d^n / n, d = 2.69832910647421123126399866618837... (see A246828). - Vaclav Kotesovec, Aug 25 2014

Extensions

More terms from Alois P. Heinz, Sep 09 2008