A305853 Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

1, 3, 10, 62, 446, 3975, 41098, 484152, 6390488, 93419965, 1498268466, 26159940522, 494036061550, 10035451747919, 218207845446062, 5057251219752612, 124462048466812950, 3241773988594489244, 89093816361187396674, 2576652694087236419386, 78224564280680539732266

Offset: 1

Author

Alois P. Heinz, Jun 11 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..424

Crossrefs

Cf. A000670, A095993, A305846, A305852.

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(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[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[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
a /@ Range[1, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000670(n) * x^n.

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