A288785 Number of blocks of size >= three in all set partitions of n.

1, 5, 26, 137, 750, 4307, 25996, 164825, 1096217, 7633650, 55549664, 421599778, 3331027887, 27349472297, 232967157736, 2055635993935, 18762063976810, 176896220650029, 1720762736285790, 17249873608817569, 178010337967774511, 1889129778601708612

Offset: 3

Author

Alois P. Heinz, Jun 15 2017

Keywords

• nonn

Examples

			a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
a(5) = 26: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
a(6) = 137: 123456, 12345|6, 12346|5, ..., 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234, ..., 1|256|3|4, 1|2|356|4, 1|2|3|456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..575
• Wikipedia, Partition of a set

Crossrefs

Column k=3 of A283424.

Cf. A000110, A355144.

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, 3):
seq(a(n), n=3..30);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,
     `if`(j>2, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=3..30);  # Alois P. Heinz, Jan 06 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, 3];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

Formula

a(n) = Bell(n+1) - Sum_{j=0..2} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-3} binomial(n,j) * Bell(j).

a(n) = Sum_{k=1..n} k * A355144(n,k). - Alois P. Heinz, Jun 20 2022

E.g.f.: (exp(x) - 1 - x - x^2/2) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 24 2022