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%I #13 Nov 21 2022 08:26:17
%S 1,1,2,10,59,446,3965,41098,484090,6390488,93419519,1498268466,
%T 26159936547,494036061550,10035451706821,218207845446062,
%U 5057251219268460,124462048466812950,3241773988588098756,89093816361187396674,2576652694087142999421
%N Inverse Euler transform of the ordered Bell numbers A000670.
%H Alois P. Heinz, <a href="/A095993/b095993.txt">Table of n, a(n) for n = 0..423</a>
%F Product(1/(1-q^n)^(a(n)), n >=1) = sum(A000670(k)*q^k, k>=0).
%F a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Oct 09 2019
%p 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(%);
%p # The function EulerInvTransform is defined in A358451.
%p a := EulerInvTransform(A000670):
%p seq(a(n), n = 0..22); # _Peter Luschny_, Nov 21 2022
%t 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 *)
%Y Cf. A000670, A085686, A095989.
%K nonn
%O 0,3
%A _Mike Zabrocki_, Jul 18 2004
%E a(0)=1 inserted by _Alois P. Heinz_, Feb 20 2017
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