A363429 Number of set partitions of [n] such that each block has at most one even element.

1, 1, 2, 5, 10, 37, 77, 372, 799, 4736, 10427, 73013, 163967, 1322035, 3017562, 27499083, 63625324, 646147067, 1512354975, 16926317722, 40012800675, 489109544320, 1166271373797, 15455199988077, 37134022033885, 530149003318273, 1282405154139046, 19619325078384593

Offset: 0

Author

Alois P. Heinz, Jun 01 2023

Keywords

• nonn

Examples

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
a(4) = 10: 123|4, 12|34, 12|3|4, 134|2, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..772
• Wikipedia, Partition of a set

Crossrefs

Bisection gives: A134980 (even part).

Cf. A000110, A110132 (exactly one even), A124421 (at least one even), A363430 (at most one odd).

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=0..ceiling(n/2)} floor(n/2)^k * binomial(ceiling(n/2),k) * Bell(ceiling(n/2)-k).