A288790 Number of blocks of size >= eight in all set partitions of n.

1, 10, 101, 947, 8670, 79249, 730745, 6838642, 65197797, 634656360, 6316333291, 64318009411, 670336612614, 7151290120037, 78085166445577, 872478836270306, 9972817907218608, 116575837400037486, 1393037460835481622, 17010118386233081680, 212160149063581345610

Offset

8,2

Links

Alois P. Heinz, Table of n, a(n) for n = 8..575

Wikipedia, Partition of a set

Formula

a(n) = Bell(n+1) - Sum_{j=0..7} binomial(n,j) * Bell(n-j).

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

E.g.f.: (exp(x) - Sum_{k=0..7} 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, 8):
seq(a(n), n=8..30);

Mathematica

Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 8}], {n, 8, 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 - 7))
print([a(n) for n in range(8, 31)]) # Indranil Ghosh, Jul 06 2017

Crossrefs

Column k=8 of A283424.

Cf. A000110.

Sequence in context: A214390 A293804 A210167 * A280371 A105032 A281102

Adjacent sequences: A288787 A288788 A288789 * A288791 A288792 A288793

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2017

Status

approved