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

0,3

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..466

Wikipedia, Partition of a set

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A121207, A124496, A276961, A297926.

Sequence in context: A002793 A162509 A371524 * A151341 A349603 A285868

Adjacent sequences: A297921 A297922 A297923 * A297925 A297926 A297927

Keyword

nonn

Author

Alois P. Heinz, Jan 08 2018

Status

approved