A374320 Number of partitions of [n] such that the number of blocks of size k is a multiple of k for every k.

1, 1, 1, 1, 4, 16, 46, 106, 316, 1604, 8156, 33716, 125456, 1073216, 10233224, 69873896, 364469561, 2296961801, 19124734801, 147200743489, 960313414036, 6422446261456, 52845891370966, 461844834503746, 3779922654292324, 31131912140021452, 296987899271509252

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: 1|2.
a(3) = 1: 1|2|3.
a(4) = 4: 12|34, 13|24, 14|23, 1|2|3|4.
a(5) = 16: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34, 1|2|3|4|5.
a(9) = 1604: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 1|2|3|49|5|6|78, 1|2|3|4|59|6|78, 1|2|3|4|5|69|78, 1|2|3|4|5|6|7|8|9.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..579
• Wikipedia, Partition of a set

Crossrefs

Cf. A374262, A374319, A374321, A374329.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(combinat[multinomial](n, i$i*j, n-i^2*j)*
      b(n-i^2*j, i-1)/(i*j)!, j=0..n/i^2))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..28);