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A305846 Inverse Weigh transform of the Bell numbers (A000110).

%I A305846 #22 Mar 10 2022 09:28:46
%S A305846 1,2,3,11,34,138,610,2976,15612,87905,526274,3334988,22270254,
%T A305846 156173299,1146640394,8791427525,70227355786,583283756678,
%U A305846 5027823752930,44903579714037,414877600876638,3959945233249877,38996757506464858,395749369601741015,4134132167178705732
%N A305846 Inverse Weigh transform of the Bell numbers (A000110).
%H A305846 Alois P. Heinz, <a href="/A305846/b305846.txt">Table of n, a(n) for n = 1..576</a>
%F A305846 Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} Bell(n) * x^n.
%p A305846 g:= proc(n) option remember; `if`(n=0, 1,
%p A305846       add(binomial(n-1, j-1)*g(n-j), j=1..n))
%p A305846     end:
%p A305846 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305846       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A305846     end:
%p A305846 a:= proc(n) option remember; g(n)-b(n, n-1) end:
%p A305846 seq(a(n), n=1..30);
%t A305846 g[n_] := g[n] = If[n == 0, 1,
%t A305846      Sum[Binomial[n-1, j-1]*g[n-j], {j, 1, n}]];
%t A305846 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A305846      Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
%t A305846 a[n_] := a[n] = g[n] - b[n, n - 1];
%t A305846 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Mar 10 2022, after _Alois P. Heinz_ *)
%Y A305846 Cf. A000110, A085686, A168246, A305850, A305853.
%K A305846 nonn
%O A305846 1,2
%A A305846 _Alois P. Heinz_, Jun 11 2018