A297924 Number of set partitions of [2n] in which the size of the last block is n.

1, 1, 4, 20, 125, 952, 8494, 86025, 969862, 12020580, 162203607, 2363458396, 36930606254, 615302885459, 10878670826170, 203268056115256, 3999642836434361, 82617423216826640, 1786559190116778030, 40344863179696283037, 949348461372003462390

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) = 4: 12|34, 13|24, 14|23, 1|2|34.
a(3) = 20: 123|456, 124|356, 125|346, 126|345, 12|3|456, 134|256, 135|246, 136|245, 13|2|456, 145|236, 146|235, 156|234, 1|23|456, 14|2|356, 1|24|356, 15|2|346, 1|25|346, 16|2|345, 1|26|345, 1|2|3|456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..466
• Wikipedia, Partition of a set

Crossrefs

Cf. A121207, A124496, A276961, A297926.

Programs

Maple

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

Mathematica

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

Formula

a(n) = A121207(2n,n) = A124496(2n,n).