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

1, 2, 3, 11, 34, 138, 610, 2976, 15612, 87905, 526274, 3334988, 22270254, 156173299, 1146640394, 8791427525, 70227355786, 583283756678, 5027823752930, 44903579714037, 414877600876638, 3959945233249877, 38996757506464858, 395749369601741015, 4134132167178705732

Offset: 1

Author

Alois P. Heinz, Jun 11 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..576

Crossrefs

Cf. A000110, A085686, A168246, A305850, A305853.

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(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..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[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)

Formula

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} Bell(n) * x^n.