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

0,3

Comments

The blocks are ordered with increasing least elements.

a(0) = 1 by convention.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..445

Wikipedia, Partition of a set

Formula

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A005416 A300989 A105617 * A094072 A361193 A363315

Adjacent sequences: A297923 A297924 A297925 * A297927 A297928 A297929

Keyword

nonn

Author

Alois P. Heinz, Jan 08 2018

Status

approved