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A363430
Number of set partitions of [n] such that each block has at most one odd element.
2
1, 1, 2, 3, 10, 17, 77, 141, 799, 1540, 10427, 20878, 163967, 338233, 3017562, 6376149, 63625324, 137144475, 1512354975, 3315122947, 40012800675, 88981537570, 1166271373797, 2626214876310, 37134022033885, 84540738911653, 1282405154139046, 2948058074576995
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/2)} ceiling(n/2)^k * binomial(floor(n/2),k) * Bell(floor(n/2)-k).
EXAMPLE
a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 3: 12|3, 1|23, 1|2|3.
a(4) = 10: 124|3, 12|34, 12|3|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 17: 124|3|5, 12|34|5, 12|3|45, 12|3|4|5, 14|23|5, 1|234|5, 1|23|45, 1|23|4|5, 14|25|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|25|34, 1|2|34|5, 1|25|3|4, 1|2|3|45, 1|2|3|4|5.
MAPLE
b:= proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m+1)+m*b(n-1, m))
end:
a:= n-> (h-> b(h, n-h))(iquo(n, 2)):
seq(a(n), n=0..30);
CROSSREFS
Bisection gives: A134980 (even part).
Cf. A000110, A110138 (exactly one odd), A124423 (at least one odd), A363429 (at most one even).
Sequence in context: A300285 A192798 A143609 * A350913 A066915 A339924
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 01 2023
STATUS
approved