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

Author

Alois P. Heinz, Jun 15 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..575
• Wikipedia, Partition of a set

Crossrefs

Column k=9 of A283424.

Cf. A000110.

Programs

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 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 *)

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

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