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A305850 Weigh transform of the Bell numbers (A000110).

%I A305850 #16 Mar 12 2022 11:33:26
%S A305850 1,1,2,7,21,78,305,1304,6007,29854,159012,904986,5479078,35150263,
%T A305850 238033523,1695554145,12663533586,98881246850,805128085616,
%U A305850 6820302066048,59983405937707,546690232627480,5154757226832625,50208266917662433,504482106565647708
%N A305850 Weigh transform of the Bell numbers (A000110).
%H A305850 Alois P. Heinz, <a href="/A305850/b305850.txt">Table of n, a(n) for n = 0..576</a>
%F A305850 G.f.: Product_{k>=1} (1+x^k)^Bell(k).
%p A305850 g:= proc(n) option remember; `if`(n=0, 1,
%p A305850       add(binomial(n-1, j-1)*g(n-j), j=1..n))
%p A305850     end:
%p A305850 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305850       add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A305850     end:
%p A305850 a:= n-> b(n$2):
%p A305850 seq(a(n), n=0..30);
%t A305850 g[n_] := g[n] = If[n == 0, 1,
%t A305850      Sum[Binomial[n - 1, j - 1]*g[n - j], {j, 1, n}]];
%t A305850 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A305850      Sum[Binomial[g[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
%t A305850 a[n_] := b[n, n];
%t A305850 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 12 2022, after _Alois P. Heinz_ *)
%Y A305850 Cf. A000110, A290351, A305846, A305852.
%K A305850 nonn
%O A305850 0,3
%A A305850 _Alois P. Heinz_, Jun 11 2018