A290352 Euler transform of the Fubini numbers (ordered Bell numbers, A000670).

1, 1, 4, 17, 98, 678, 5687, 55656, 626161, 7963511, 113027113, 1770785023, 30346490633, 564546034917, 11327726548719, 243811768229012, 5602495216123312, 136878883607160468, 3542830077444873188, 96835203745704714722, 2787051847418347608600

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A000670, A007003, A095993, A290351.

Programs

Maple

b:= proc(n, m) option remember;
     `if`(n=0, m!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d, 0), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);

Mathematica

b[n_, m_]:=b[n, m]=If[n==0, m!, Sum[b[n - 1, Max[m, j]], {j, m + 1}]]; a[n_]:=a[n]=If[n==0, 1, Sum[Sum[d*b[d, 0], {d, Divisors[j]}] a[n - j], {j, n}]/n]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Jul 28 2017, after Maple code *)

Formula

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, May 31 2019