A085686 Inverse Euler transform of Bell numbers.

1, 1, 3, 9, 34, 135, 610, 2965, 15612, 87871, 526274, 3334850, 22270254, 156172689, 1146640394, 8791424549, 70227355786, 583283741066, 5027823752930, 44903579626132, 414877600876638, 3959945232723603, 38996757506464858, 395749369598406027, 4134132167178705732

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..300

Maple

read transforms; A := series(exp(exp(x)-1), x, 60); A000110 := n->n!*coeff(A, x, n); [seq(A000110(i), i=1..30)]; EULERi(%);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(combinat:-bell):
seq(a(n), n = 1..25); # Peter Luschny, Nov 21 2022

Mathematica

n=24; eq[0] = Rest[ Thread[ CoefficientList[ 1 + Series[ Sum[ BellB[k]*x^k, {k, 1, n}] - Product[1/(1-x^k)^a[k], {k, 1, n}], {x, 0, n}], x] == 0]]; s[1] = First[ Solve[ First[eq[0]], a[1]]]; Do[eq[k] = Rest[eq[k-1]] /. s[k]; s[k+1] = First[ Solve[ First[eq[k]], a[k+1]]], {k, 1, n-1}]; Table[a[k], {k, 1, n}] /. Flatten[Table[s[k], {k, 1, n}]] (* Jean-François Alcover, Jul 26 2011 *)
bb = Array[BellB, n = 25]; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i* bb[[i]] - Sum[s[[d]]*bb[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)

Crossrefs

Cf. A000110, A305846.

Sequence in context: A137953 A353944 A245893 * A191412 A371542 A349017

Adjacent sequences: A085683 A085684 A085685 * A085687 A085688 A085689

Keyword

nonn

Author

N. J. A. Sloane, Jul 18 2003

Status

approved