A288787 Number of blocks of size >= five in all set partitions of n.

1, 7, 50, 345, 2392, 16955, 123707, 932010, 7260709, 58509323, 487593202, 4199841037, 37361858716, 342989895895, 3246458915947, 31653980371254, 317654338317380, 3278058775976704, 34757921507150964, 378372365291381716, 4225533329681577846, 48375204740642752562

Offset

5,2

Links

Alois P. Heinz, Table of n, a(n) for n = 5..575

Wikipedia, Partition of a set

Formula

a(n) = Bell(n+1) - Sum_{j=0..4} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-5} binomial(n,j) * Bell(j).

E.g.f.: (exp(x) - Sum_{k=0..4} 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, 5):
seq(a(n), n=5..30);

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, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

Crossrefs

Column k=5 of A283424.

Cf. A000110.

Sequence in context: A227676 A278875 A266360 * A033117 A096882 A033125

Adjacent sequences: A288784 A288785 A288786 * A288788 A288789 A288790

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2017

Status

approved