A288789 Number of blocks of size >= 7 in all set partitions of n.

1, 9, 82, 701, 5897, 49854, 427597, 3740609, 33479542, 307119477, 2890138160, 27911144971, 276632735047, 2813333368854, 29349063282197, 313940448544057, 3441759044602385, 38652680805862224, 444450158120668786, 5229815283321976222, 62942722623990478840

Offset

7,2

Links

Alois P. Heinz, Table of n, a(n) for n = 7..575

Wikipedia, Partition of a set

Formula

a(n) = Bell(n+1) - Sum_{j=0..6} binomial(n,j) * Bell(n-j).

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

E.g.f.: (exp(x) - Sum_{k=0..6} 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, 7):
seq(a(n), n=7..30);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
      `if`(j>6, [0, p[1]], 0))(b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=7..30);  # Alois P. Heinz, Jun 26 2022

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k + 1] + Binomial[n, k]*b[n - k]];
a[n_] := g[n, 7];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

Crossrefs

Column k=7 of A283424.

Cf. A000110.

Sequence in context: A275917 A293803 A263817 * A033119 A033127 A361715

Adjacent sequences: A288786 A288787 A288788 * A288790 A288791 A288792

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2017

Status

approved