A110132
a(n) = floor(n/2)^ceiling(n/2).
Original entry on oeis.org
1, 0, 1, 1, 4, 8, 27, 81, 256, 1024, 3125, 15625, 46656, 279936, 823543, 5764801, 16777216, 134217728, 387420489, 3486784401, 10000000000, 100000000000, 285311670611, 3138428376721, 8916100448256, 106993205379072, 302875106592253, 3937376385699289
Offset: 0
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Join[{1},Table[Floor[n/2]^Ceiling[n/2],{n,30}]] (* Harvey P. Dale, Feb 16 2025 *)
A231601
Number of permutations of [n] avoiding ascents from odd to even numbers.
Original entry on oeis.org
1, 1, 1, 4, 8, 54, 162, 1536, 6144, 75000, 375000, 5598720, 33592320, 592950960, 4150656720, 84557168640, 676457349120, 15620794116480, 140587147048320, 3628800000000000, 36288000000000000, 1035338990313196800, 11388728893445164800, 355902198372945100800
Offset: 0
a(0) = 1: ().
a(1) = 1: 1.
a(2) = 1: 21.
a(3) = 4: 132, 213, 231, 321.
a(4) = 8: 1324, 2413, 2431, 3241, 4132, 4213, 4231, 4321.
a(5) = 54: 13245, 13254, 13524, ..., 54213, 54231, 54321.
a(6) = 162: 132465, 132546, 132645, ..., 654213, 654231, 654321.
Bisection gives:
A061711 (even part).
A363430
Number of set partitions of [n] such that each block has at most one odd element.
Original entry on oeis.org
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
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.
Bisection gives:
A134980 (even part).
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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);
Showing 1-3 of 3 results.
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