A276961 Number of set partitions of [2n] with largest set of size n.

1, 1, 9, 90, 1015, 12978, 187110, 3008148, 53275365, 1028142830, 21426984722, 478684639524, 11394222257054, 287518726261900, 7658231720886900, 214521099685649640, 6299407928673657135, 193373975592937777770, 6189939300880260745050, 206159811915115686404700

Offset: 0

Author

Alois P. Heinz, Sep 22 2016

Keywords

• nonn

Comments

The blocks are ordered with increasing least elements.

a(0) = 1 by convention.

Examples

			a(1) = 1: 1|2.
a(2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
		

Links

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

Crossrefs

Cf. A000110, A080510, A276923, A297924, A297926, A327884, A328156.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
a:= n-> `if`(n=0, 1, b(2*n, n)-b(2*n, n-1)):
seq(a(n), n=0..20);

Mathematica

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

Formula

a(n) = A080510(2n,n).

a(n) = A327884(2n,n).

a(n) = ceiling(C(2n,n)*(A000110(n)-1/2)). - Ludovic Schwob, Jan 15 2022