A305850 Weigh transform of the Bell numbers (A000110).

1, 1, 2, 7, 21, 78, 305, 1304, 6007, 29854, 159012, 904986, 5479078, 35150263, 238033523, 1695554145, 12663533586, 98881246850, 805128085616, 6820302066048, 59983405937707, 546690232627480, 5154757226832625, 50208266917662433, 504482106565647708

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Formula

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

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(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[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[g[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000110, A290351, A305846, A305852.

Sequence in context: A339041 A319852 A060900 * A151289 A150300 A150301

Adjacent sequences: A305847 A305848 A305849 * A305851 A305852 A305853

Keyword

nonn

Author

Alois P. Heinz, Jun 11 2018

Status

approved