A288792 Number of blocks of size >= ten in all set partitions of n.

1, 12, 145, 1600, 17032, 179132, 1883117, 19929390, 213332101, 2316793121, 25577181324, 287421068697, 3290394397097, 38393883291996, 456753452800691, 5540597439008861, 68530489547341697, 864218608315007230, 11109867095322262250, 145563654356205885737

Offset

10,2

Links

Alois P. Heinz, Table of n, a(n) for n = 10..576

Wikipedia, Partition of a set

Formula

a(n) = Bell(n+1) - Sum_{j=0..9} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-10} binomial(n,j) * Bell(j).

E.g.f.: (exp(x) - Sum_{k=0..9} 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, 10):
seq(a(n), n=10..30);

Mathematica

Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 10}], {n, 10, 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 - 9)])
print([a(n) for n in range(10, 31)]) # Indranil Ghosh, Jul 06 2017

Crossrefs

Column k=10 of A283424.

Cf. A000110.

Sequence in context: A067219 A075619 A055332 * A041061 A174227 A041266

Adjacent sequences: A288789 A288790 A288791 * A288793 A288794 A288795

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2017

Status

approved