A350648 Sum over all partitions of [n] of the number of blocks containing their own index when blocks are ordered with decreasing largest elements.

0, 1, 1, 5, 11, 48, 173, 795, 3719, 19343, 106563, 628508, 3923602, 25875858, 179468739, 1305268102, 9925892324, 78728325373, 649856661196, 5571421770478, 49521735963376, 455616186779543, 4332419124871058, 42520560822961111, 430191406640367880

Offset: 0

Author

Alois P. Heinz, Jan 09 2022

Keywords

• nonn

Examples

			a(3) = 5 = 3*1 + 2*2: 321, 3|21, 3|2|1; 31|2.
a(4) = 11 = 7*1 + 2*2: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2; 431|2, 41|32.
		

Links

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

Crossrefs

Cf. A350589, A350647.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p->p+
      [0, `if`(j=n, p[1], 0)])(b(n-1, max(j, m))), j=1..m+1))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[j == n, p[[1]], 0]}][b[n - 1, Max[j, m]]], {j, 1, m + 1}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..ceiling(n/2)} k * A350647(n,k).