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A305853 Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

%I A305853 #18 Dec 21 2020 11:54:03
%S A305853 1,3,10,62,446,3975,41098,484152,6390488,93419965,1498268466,
%T A305853 26159940522,494036061550,10035451747919,218207845446062,
%U A305853 5057251219752612,124462048466812950,3241773988594489244,89093816361187396674,2576652694087236419386,78224564280680539732266
%N A305853 Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).
%H A305853 Alois P. Heinz, <a href="/A305853/b305853.txt">Table of n, a(n) for n = 1..424</a>
%F A305853 Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000670(n) * x^n.
%F A305853 a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Sep 10 2019
%p A305853 g:= proc(n) option remember; `if`(n=0, 1,
%p A305853       add(g(n-j)*binomial(n, j), j=1..n))
%p A305853     end:
%p A305853 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305853       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A305853     end:
%p A305853 a:= proc(n) option remember; g(n)-b(n, n-1) end:
%p A305853 seq(a(n), n=1..30);
%t A305853 g[n_] := g[n] = If[n == 0, 1,
%t A305853     Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
%t A305853 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A305853     Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
%t A305853 a[n_] := a[n] = g[n] - b[n, n - 1];
%t A305853 a /@ Range[1, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y A305853 Cf. A000670, A095993, A305846, A305852.
%K A305853 nonn
%O A305853 1,2
%A A305853 _Alois P. Heinz_, Jun 11 2018