A110132 -id:A110132 - cp's OEIS frontend

A110138 a(n) = ceiling(n/2)^floor(n/2).

1, 1, 1, 2, 4, 9, 27, 64, 256, 625, 3125, 7776, 46656, 117649, 823543, 2097152, 16777216, 43046721, 387420489, 1000000000, 10000000000, 25937424601, 285311670611, 743008370688, 8916100448256, 23298085122481, 302875106592253, 793714773254144, 11112006825558016

Offset: 0

Paul Barry, Jul 13 2005

a(n) is the number of partitions of [n] such that each block has exactly one odd element: a(5) = 9: 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34. - Alois P. Heinz, Jun 01 2023

Alois P. Heinz, Table of n, a(n) for n = 0..773

Cf. A000312, A110132.

def A110138(n): return ((m:=n>>1)+(n&1))**m # Chai Wah Wu, Jan 18 2023

a(2n) = A110132(2n) = A000312(n).

A363429 Number of set partitions of [n] such that each block has at most one even element.

Original entry on oeis.org

1, 1, 2, 5, 10, 37, 77, 372, 799, 4736, 10427, 73013, 163967, 1322035, 3017562, 27499083, 63625324, 646147067, 1512354975, 16926317722, 40012800675, 489109544320, 1166271373797, 15455199988077, 37134022033885, 530149003318273, 1282405154139046, 19619325078384593

Offset: 0

Alois P. Heinz, Jun 01 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
a(4) = 10: 123|4, 12|34, 12|3|4, 134|2, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..772
Wikipedia, Partition of a set

Bisection gives: A134980 (even part).

Cf. A000110, A110132 (exactly one even), A124421 (at least one even), A363430 (at most one odd).

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(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..30);

a(n) = Sum_{k=0..ceiling(n/2)} floor(n/2)^k * binomial(ceiling(n/2),k) * Bell(ceiling(n/2)-k).

cp's OEIS Frontend

A110138 a(n) = ceiling(n/2)^floor(n/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula