A363430 Number of set partitions of [n] such that each block has at most one odd element.

1, 1, 2, 3, 10, 17, 77, 141, 799, 1540, 10427, 20878, 163967, 338233, 3017562, 6376149, 63625324, 137144475, 1512354975, 3315122947, 40012800675, 88981537570, 1166271373797, 2626214876310, 37134022033885, 84540738911653, 1282405154139046, 2948058074576995

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) = 3: 12|3, 1|23, 1|2|3.
a(4) = 10: 124|3, 12|34, 12|3|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 17: 124|3|5, 12|34|5, 12|3|45, 12|3|4|5, 14|23|5, 1|234|5, 1|23|45, 1|23|4|5, 14|25|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|25|34, 1|2|34|5, 1|25|3|4, 1|2|3|45, 1|2|3|4|5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..773
• Wikipedia, Partition of a set

Crossrefs

Bisection gives: A134980 (even part).

Cf. A000110, A110138 (exactly one odd), A124423 (at least one odd), A363429 (at most one even).

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(h, n-h))(iquo(n, 2)):
seq(a(n), n=0..30);

Formula

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