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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..598

Wikipedia, Partition of a set

Example

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.

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);

Crossrefs

Cf. A374292, A374320, A374321, A374329.

Sequence in context: A082595 A262154 A080072 * A149091 A149092 A188117

Adjacent sequences: A374316 A374317 A374318 * A374320 A374321 A374322

Keyword

nonn

Author

Alois P. Heinz, Jul 04 2024

Status

approved