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

1,2

Links

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

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

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 *)

Crossrefs

Cf. A000670, A095993, A305846, A305852.

Sequence in context: A228773 A034889 A260969 * A160921 A042705 A041014

Adjacent sequences: A305850 A305851 A305852 * A305854 A305855 A305856

Keyword

nonn

Author

Alois P. Heinz, Jun 11 2018

Status

approved