A305852 Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

1, 1, 3, 16, 91, 658, 5567, 54917, 620081, 7905592, 112382245, 1762646331, 30231516786, 562750751610, 11297034281595, 243241826522376, 5591075279423398, 136633359995403580, 3537193288612096901, 96697587673174195740, 2783492094736121087958

Offset: 0

Author

Alois P. Heinz, Jun 11 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..424

Crossrefs

Cf. A000670, A290352, A305850, A305853.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(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[g[n - j] Binomial[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];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

G.f.: Product_{k>=1} (1+x^k)^A000670(k).

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 10 2019