A372649 Total sum over all partitions of [n] of the number of maximal blocks.

0, 1, 3, 7, 21, 71, 293, 1268, 6107, 31123, 170745, 998966, 6212627, 40854360, 283290348, 2059884614, 15667307457, 124266461587, 1025342179759, 8784261413616, 78003593175261, 716854898767936, 6808817431686858, 66754426111124686, 674754718441688851

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Wikipedia, Partition of a set

Formula

a(n) = Sum_{k=0..n} k * A372722(n,k).

Example

a(3) = 7 = 3 + 1 + 1 + 1 + 1: 1|2|3, 1|23, 12|3, 13|2, 123.
a(4) = 21 = 1+1+1+2+1+1+2+1+2+1+1+1+1+1+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.

Maple

b:= proc(n, m, t) option remember; `if`(n=0, t,
      add(binomial(n-1, j-1)*b(n-j, max(j, m),
     `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
    end:
a:= n-> b(n, 0$2):
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, Max[j, m],
   If[j > m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 10 2024, after Alois P. Heinz *)

Crossrefs

Cf. A000110, A097148, A372650, A372722.

Sequence in context: A105795 A322459 A244174 * A148678 A148679 A148680

Adjacent sequences: A372646 A372647 A372648 * A372650 A372651 A372652

Keyword

nonn

Author

Alois P. Heinz, May 08 2024

Status

approved