A374319 Number of partitions of [n] such that the number of blocks of size k is zero or a divisor of k for every k.

1, 1, 1, 4, 8, 31, 82, 274, 1626, 5135, 26751, 125489, 1020692, 4333707, 31083613, 132960104, 1323145731, 8282668312, 70017330978, 423293287673, 3135764479898, 30762429056580, 269133472001923, 2185746568531948, 15121514389566421, 147045774699171957

Offset: 0

Author

Alois P. Heinz, Jul 04 2024

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 8: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234.
a(5) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..598
• Wikipedia, Partition of a set

Crossrefs

Cf. A374292, A374320, A374321, A374329.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(i, j)=0, b(n-i*j, i-1)/j!*
      combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..27);