cp's OEIS Frontend

A085686 Inverse Euler transform of Bell numbers.

%I A085686 #27 May 16 2025 00:56:13
%S A085686 1,1,3,9,34,135,610,2965,15612,87871,526274,3334850,22270254,
%T A085686 156172689,1146640394,8791424549,70227355786,583283741066,
%U A085686 5027823752930,44903579626132,414877600876638,3959945232723603,38996757506464858,395749369598406027,4134132167178705732
%N A085686 Inverse Euler transform of Bell numbers.
%H A085686 Alois P. Heinz, <a href="/A085686/b085686.txt">Table of n, a(n) for n = 1..300</a>
%H A085686 Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, and Mike Zabrocki, <a href="https://arxiv.org/abs/2505.06941">When are Hopf algebras determined by integer sequences?</a>, arXiv:2505.06941 [math.CO], 2025. See p. 3.
%p A085686 read transforms; A := series(exp(exp(x)-1),x,60); A000110 := n->n!*coeff(A,x,n); [seq(A000110(i),i=1..30)]; EULERi(%);
%p A085686 # The function EulerInvTransform is defined in A358451.
%p A085686 a := EulerInvTransform(combinat:-bell):
%p A085686 seq(a(n), n = 1..25); # _Peter Luschny_, Nov 21 2022
%t A085686 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 *)
%t A085686 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 *)
%Y A085686 Cf. A000110, A305846.
%K A085686 nonn
%O A085686 1,3
%A A085686 _N. J. A. Sloane_, Jul 18 2003