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

Author

Alois P. Heinz, Jun 15 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 7..575
• Wikipedia, Partition of a set

Crossrefs

Column k=7 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, 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 *)

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