A275282 Number of set partitions of [n] with symmetric block size list.

1, 1, 2, 2, 7, 9, 47, 80, 492, 985, 7197, 16430, 139316, 361737, 3425683, 9939134, 103484333, 329541459, 3747921857, 12980700318, 159811532315, 598410986533, 7902918548186, 31781977111506, 447462660895105, 1920559118957107, 28699615818386524, 130838216971937408

Offset: 0

Author

Alois P. Heinz, Jul 21 2016

Keywords

• nonn

Examples

			a(3) = 2: 123, 1|2|3.
a(4) = 7: 1234, 12|34, 13|24, 14|23, 1|23|4, 1|24|3, 1|2|3|4.
a(5) = 9: 12345, 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34, 1|2|3|4|5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Wikipedia, Partition of a set

Crossrefs

Row sums of A275281.

Programs

Maple

b:= proc(n, s) option remember; `if`(n>s,
      binomial(n-1, n-s-1), 1) +add(binomial(n-1, j-1)*
      b(n-j, s+j) *binomial(s+j-1, j-1), j=1..(n-s)/2)
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);

Mathematica

b[n_, s_] := b[n, s] = If[n > s, Binomial[n-1, n-s-1], 1] + Sum[Binomial[n - 1, j - 1]*b[n - j, s + j]*Binomial[s + j - 1, j - 1], {j, 1, (n-s)/2}];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2018, from Maple *)

Formula

a(n) = Sum_{k=0..n} A275281(n,k).