A288791 Number of blocks of size >= nine in all set partitions of n.

1, 11, 122, 1245, 12325, 121136, 1195147, 11915997, 120572790, 1241499241, 13030331671, 139549798524, 1525923634907, 17041290249637, 194394900237176, 2264977282222371, 26951265841776186, 327445918493429897, 4060993235341162405, 51396034231430455550

Offset

9,2

Links

Alois P. Heinz, Table of n, a(n) for n = 9..575

Wikipedia, Partition of a set

Formula

a(n) = Bell(n+1) - Sum_{j=0..8} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-9} binomial(n,j) * Bell(j).

E.g.f.: (exp(x) - Sum_{k=0..8} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
g:= proc(n, k) option remember; `if`(n<k, 0,
      g(n, k+1) +binomial(n, k)*b(n-k))
    end:
a:= n-> g(n, 9):
seq(a(n), n=9..30);

Mathematica

Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 9}], {n, 9, 30}] (* Indranil Ghosh, Jul 06 2017 *)

Prog

(Python)
from sympy import bell, binomial
def a(n): return sum([binomial(n, j)*bell(j) for j in range(n - 8)])
print([a(n) for n in range(9, 31)]) # Indranil Ghosh, Jul 06 2017

Crossrefs

Column k=9 of A283424.

Cf. A000110.

Sequence in context: A368114 A293805 A358697 * A049666 A163462 A334000

Adjacent sequences: A288788 A288789 A288790 * A288792 A288793 A288794

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2017

Status

approved