A297926 Number of set partitions of [2n] in which the size of the first block is n.

1, 1, 6, 50, 525, 6552, 93786, 1504932, 26640900, 514083570, 10713538550, 239342496120, 5697111804566, 143759365731100, 3829115870472600, 107260549881604200, 3149703964487098665, 96686987797052290440, 3094969650442399156350, 103079905957566679518300

Offset: 0

Author

Alois P. Heinz, Jan 08 2018

Keywords

• nonn

Comments

The blocks are ordered with increasing least elements.

a(0) = 1 by convention.

Examples

			a(1) = 1: 1|2.
a(2) = 6: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 14|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..445
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A056857, A056860, A276961, A297924.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> binomial(2*n-1, n-1)*b(n):
seq(a(n), n=0..25);

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := Binomial[2*n-1, n-1] * b[n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

a(n) = binomial(2*n-1,n-1) * Bell(n).

a(n) = A056857(2n,n) = A056860(2n,n).