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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..579

Wikipedia, Partition of a set

Example

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.

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);

Crossrefs

Cf. A374262, A374319, A374321, A374329.

Sequence in context: A159940 A379464 A000704 * A374262 A007315 A055342

Adjacent sequences: A374317 A374318 A374319 * A374321 A374322 A374323

Keyword

nonn

Author

Alois P. Heinz, Jul 04 2024

Status

approved