A374329 Number of partitions of [n] such that the number of blocks of size k is a multiple or a divisor of k for every k.

1, 1, 2, 5, 15, 52, 178, 702, 3160, 15267, 72904, 401139, 2473320, 15932022, 99025826, 641593206, 4570555035, 35589342752, 281312386194, 2198910615580, 17652574088975, 154057648160882, 1437155645037380, 13320654235409156, 121511456568711443, 1135554227462348853

Offset: 0

Author

Alois P. Heinz, Jul 04 2024

Keywords

• nonn

Examples

			a(6) = 178 = 203 - 25 counts all partitions of [6] with the exception of 123|456, 124|356, 125|346, 126|345, 12|34|56, 12|35|46, 12|36|45, 134|256, 135|246, 136|245, 13|24|56, 13|25|46, 13|26|45, 145|236, 146|235, 14|23|56, 156|234, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..578
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A372545, A374319, A374320, A374321.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(
      `if`(irem(j, i)=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..25);