cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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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Author

Paul Barry, Jul 13 2005

Keywords

Comments

Central coefficients T(n,floor(n/2)) of number triangle A004248. The central coefficients T(2n,n) of this triangle are n^n or A000312.
a(n) is the number of partitions of [n] such that each block has exactly one even element: a(5) = 8: 1235|4, 123|45, 125|34, 12|345, 1345|2, 134|25, 145|23, 14|235. - Alois P. Heinz, Jun 01 2023

Crossrefs

Programs

  • Mathematica
    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

Views

Author

Alois P. Heinz, Nov 11 2013

Keywords

Examples

			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.
		

Crossrefs

Column k=0 of A231777.
Bisection gives: A061711 (even part).

Programs

  • Maple
    a:= n-> ceil(n/2)!*ceil(n/2)^floor(n/2):
    seq(a(n), n=0..30);

Formula

a(n) = ceiling(n/2)! * ceiling(n/2)^floor(n/2).
a(n) = A081123(n+1) * A110138(n).

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

Views

Author

Alois P. Heinz, Jun 01 2023

Keywords

Examples

			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.
		

Crossrefs

Bisection gives: A134980 (even part).
Cf. A000110, A110138 (exactly one odd), A124423 (at least one odd), A363429 (at most one even).

Programs

  • 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);

Formula

a(n) = Sum_{k=0..floor(n/2)} ceiling(n/2)^k * binomial(floor(n/2),k) * Bell(floor(n/2)-k).
Showing 1-3 of 3 results.