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A374321
Number of partitions of [n] such that the number of blocks of size k is zero or equals k for every k.
5
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
OFFSET
0,5
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 04 2024
STATUS
approved