A372650 Total sum over all partitions of [n] of the number of minimal blocks.

0, 1, 3, 7, 27, 86, 393, 1688, 8291, 44143, 248428, 1480073, 9440049, 63265606, 444309232, 3273807272, 25227429123, 202458174614, 1689474026499, 14636685675142, 131413462612012, 1220636654904548, 11712836883408675, 115956109213769404, 1182944504931376337

Offset: 0

Author

Alois P. Heinz, May 08 2024

Keywords

• nonn

Examples

			a(4) = 27 = 1+1+1+2+2+1+2+2+2+1+2+2+2+2+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A097147, A372649, A372762.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, t,
      add(binomial(n-1, j-1)*b(n-j, min(j, m),
     `if`(j b(n$2, 0):
seq(a(n), n=0..24);

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
   Sum[Binomial[n - 1, j - 1]*b[n - j, Min[j, m],
   If[j < m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 11 2024, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A372762(n,k).