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A305852 Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

%I A305852 #20 Dec 21 2020 11:55:21
%S A305852 1,1,3,16,91,658,5567,54917,620081,7905592,112382245,1762646331,
%T A305852 30231516786,562750751610,11297034281595,243241826522376,
%U A305852 5591075279423398,136633359995403580,3537193288612096901,96697587673174195740,2783492094736121087958
%N A305852 Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).
%H A305852 Alois P. Heinz, <a href="/A305852/b305852.txt">Table of n, a(n) for n = 0..424</a>
%F A305852 G.f.: Product_{k>=1} (1+x^k)^A000670(k).
%F A305852 a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Sep 10 2019
%p A305852 g:= proc(n) option remember; `if`(n=0, 1,
%p A305852       add(g(n-j)*binomial(n, j), j=1..n))
%p A305852     end:
%p A305852 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305852       add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A305852     end:
%p A305852 a:= n-> b(n$2):
%p A305852 seq(a(n), n=0..30);
%t A305852 g[n_] := g[n] = If[n == 0, 1,
%t A305852     Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
%t A305852 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0,
%t A305852     Sum[Binomial[g[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
%t A305852 a[n_] := b[n, n];
%t A305852 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y A305852 Cf. A000670, A290352, A305850, A305853.
%K A305852 nonn
%O A305852 0,3
%A A305852 _Alois P. Heinz_, Jun 11 2018