A305850 Weigh transform of the Bell numbers (A000110).

1, 1, 2, 7, 21, 78, 305, 1304, 6007, 29854, 159012, 904986, 5479078, 35150263, 238033523, 1695554145, 12663533586, 98881246850, 805128085616, 6820302066048, 59983405937707, 546690232627480, 5154757226832625, 50208266917662433, 504482106565647708

Offset: 0

Author

Alois P. Heinz, Jun 11 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576

Crossrefs

Cf. A000110, A290351, A305846, A305852.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*g(n-j), j=1..n))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Mathematica

g[n_] := g[n] = If[n == 0, 1,
     Sum[Binomial[n - 1, j - 1]*g[n - j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[Binomial[g[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

Formula

G.f.: Product_{k>=1} (1+x^k)^Bell(k).