A374321 - OEIS

A374321

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

1, 1, 0, 0, 3, 15, 0, 0, 0, 280, 2800, 0, 0, 600600, 8408400, 0, 2627625, 44669625, 0, 0, 38192529375, 802043116875, 0, 0, 0, 1508282884484376, 39215354996593776, 0, 0, 107469680368165243128, 3224090411044957293840, 0, 0, 0, 76290792475347121351680

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..607

Wikipedia, Partition of a set

FORMULA

a(n) = 0 <=> n in { A001422 }.

a(n) > 0 <=> n in { A003995 }.

EXAMPLE

a(0) = 1: the empty partition.

a(1) = 1: 1.

a(4) = 3: 12|34, 13|24, 14|23.

a(5) = 15: 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.

a(9) = 280: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 169|278|345, 178|269|345, 179|268|345, 189|267|345.

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1,

`if`(i<1, 0, add(`if`(j=0 or j=i, 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..35);

CROSSREFS

Cf. A001422, A003995, A372579, A374319, A374320, A374329.

Sequence in context: A050645 A048568 A119678 * A372579 A096682 A371099

Adjacent sequences: A374318 A374319 A374320 * A374322 A374323 A374324

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jul 04 2024

STATUS

approved