A350650 Number of partitions of [n] having exactly one block containing its own index when blocks are ordered with decreasing largest elements.

0, 1, 1, 3, 7, 25, 91, 390, 1797, 9069, 49106, 284537, 1751554, 11406588, 78254594, 563642925, 4249337018, 33443545866, 274130245342, 2335311549498, 20637538548167, 188867393030394, 1787189672368355, 17461684290203403, 175930808241047092, 1825666076751872506

Offset: 0

Author

Alois P. Heinz, Jan 09 2022

Keywords

• nonn

Examples

			a(4) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.
		

Links

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

Crossrefs

Column k=1 of A350647, A350684.

Programs

Maple

b:= proc(n, m) option remember; series(`if`(n=0, 1, add(
      `if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)), x, 2)
    end:
a:= n-> coeff(b(n, 0), x, 1):
seq(a(n), n=0..25);

Mathematica

b[n_, m_] := b[n, m] = Series[If[n == 0, 1, Sum[
    If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]], {x, 0, 2}];
a[n_] := Coefficient[b[n, 0], x, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)