A095993 Inverse Euler transform of the ordered Bell numbers A000670.

1, 1, 2, 10, 59, 446, 3965, 41098, 484090, 6390488, 93419519, 1498268466, 26159936547, 494036061550, 10035451706821, 218207845446062, 5057251219268460, 124462048466812950, 3241773988588098756, 89093816361187396674, 2576652694087142999421

Offset: 0

Author

Mike Zabrocki, Jul 18 2004

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..423

Crossrefs

Cf. A000670, A085686, A095989.

Programs

Maple

read transforms; A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n,k)*A000670(n-k),k=1..n); fi; end; [seq(A000670(i),i=1..30)]; EULERi(%);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000670):
seq(a(n), n = 0..22); # Peter Luschny, Nov 21 2022

Mathematica

max = 25; b[0] = 1; b[n_] := b[n] = Sum[Binomial[n, k]*b[n-k], {k, 1, n}]; bb = Array[b, max]; s = {}; For[i=1, i <= max, i++, AppendTo[s, i*bb[[i]] - Sum[s[[d]]*bb[[i-d]], {d, i-1}]]]; a[0] = 1; a[n_] := Sum[If[Divisible[ n, d], MoebiusMu[n/d], 0]*s[[d]], {d, 1, n}]/n; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 25 2017 *)

Formula

Product(1/(1-q^n)^(a(n)), n >=1) = sum(A000670(k)*q^k, k>=0).

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019

Extensions

a(0)=1 inserted by Alois P. Heinz, Feb 20 2017