A078707 -id:A078707 - cp's OEIS frontend

A275549 Number of classes of endofunctions of [n] under reversal.

1, 1, 3, 18, 136, 1625, 23436, 412972, 8390656, 193739769, 5000050000, 142656721086, 4458051717120, 151437584670385, 5556003465485760, 218946946471875000, 9223372039002259456, 413620131002462320337, 19673204037747448432896, 989209827833222327690890

Offset: 0

Olivier Gérard, Aug 01 2016

f and g are in the same class if function g(i) = f(n+1-i) for all i.
Decomposition by class size
.
n    1      2
---------------
1    1      0
2    2      1
3    9      9
4    16     120
5    125    1500
6    216    23220
7    2401   410571
.
Demonstration for the formula: the classes are either of size 1 or 2.
The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).
So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).
a(n) is the number of unoriented length n strings with a maximum of n colors. - Andrew Howroyd, Sep 13 2019

Andrew Howroyd, Table of n, a(n) for n = 0..200

Main diagonal of A277504.
Cf. A000312 All endofunctions
Cf. A000169 Classes under translation mod n
Cf. A001700 Classes under sort
Cf. A056665 Classes under rotation
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A081721 Classes under rotation and reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Cf. A078707 Endofunctions symmetric around their middle (stable by reversal).

a(n) = {(n^n + n^((n+1)\2))/2} \\ Andrew Howroyd, Sep 13 2019

a(n) = (n^n+n^ceiling(n/2))/2.

A152291 a(n) = (n+1)^floor((n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 5, 36, 49, 512, 729, 10000, 14641, 248832, 371293, 7529536, 11390625, 268435456, 410338673, 11019960576, 16983563041, 512000000000, 794280046581, 26559922791424, 41426511213649, 1521681143169024, 2384185791015625

Offset: 0

Paul D. Hanna, Dec 02 2008

nonn

Cf. A152290, A000272.

This is for Coxeter type A what A078707 is for Coxeter type B.

[(n+1)^((n-1) div 2): n in [0..30]]; // Vincenzo Librandi, May 31 2015

PARI
```
a(n)=(n+1)^floor((n-1)/2)
```

vector(30, n, n--; (n+1)^((n-1)\2)) \\ Michel Marcus, Jun 01 2015

Row sums of A152290 at q=-1: a(n) = Sum_{k=0..n(n-1)/2} A152290(n,k)*(-1)^k.

a(n) = denominator((-3+(-1)^n)*((1-sqrt(1+n+1/(1+n)))^n-(1+sqrt(1+n+1/(1+n)))^n)/(8*sqrt(1+n+1/(1+n)))). - Gerry Martens, May 31 2015

A092503 a(n) = n^floor(n/2).

Original entry on oeis.org

1, 1, 2, 3, 16, 25, 216, 343, 4096, 6561, 100000, 161051, 2985984, 4826809, 105413504, 170859375, 4294967296, 6975757441, 198359290368, 322687697779, 10240000000000, 16679880978201, 584318301411328, 952809757913927, 36520347436056576, 59604644775390625

Offset: 0

Reinhard Zumkeller, Apr 05 2004

nonn

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A000312 (n^n), A078707 (n^ceiling(n/2)).

a(n) = { n^(n\2) } \\ Harry J. Smith, Jun 21 2009

a(n) = n^((n-(1-(-1)^n)/2)/2). - Wesley Ivan Hurt, Mar 19 2015

A243520 Numbers that are congruent to {0, 8} mod 11.

Original entry on oeis.org

0, 8, 11, 19, 22, 30, 33, 41, 44, 52, 55, 63, 66, 74, 77, 85, 88, 96, 99, 107, 110, 118, 121, 129, 132, 140, 143, 151, 154, 162, 165, 173, 176, 184, 187, 195, 198, 206, 209, 217, 220, 228, 231, 239, 242, 250, 253, 261, 264, 272, 275, 283, 286, 294, 297, 305

Offset: 0

Viet Quoc Le Tran, Jun 14 2014

Union of A008593 and A017485. - Michel Marcus, Jun 15 2014

This sequence mimics in some sense the ceiling function of n/2 (the seq. A110654) relative to variations from a main class of recurrence relations; in order to get the ceiling function of n/2 (see Formula section), the vector v must be [0,1] instead of [3,8]. - R. J. Cano, Jun 15 2014

R. J. Cano, Table of n, a(n) for n = 0..9999
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010706, A110654, A078707.

&cat [[11*n,11*n+8]: n in [0..30]]; // [Bruno Berselli, Jun 16 2014]

A243520:=n->5*n + 2*(n mod 2) + ceil(n/2); seq(A243520(n), n=0..50); # Wesley Ivan Hurt, Jun 21 2014

Flatten[Table[11 n + {0, 8}, {n, 0, 32}]] (* Alonso del Arte, Jun 15 2014 *)

a(n)=5*n+2*(n%2)+ceil(n/2); \\ R. J. Cano, Jun 15 2014

a(n)=if(!n,0,a(n-1)+[3,8][1+n%2]); \\ R. J. Cano, Jun 15 2014

a(n) = -5/4*(-1)^n + 11*n/2 + 5/4.
From R. J. Cano, Jun 15 2014: (Start)
a(n) = 5*n + 2*(n mod 2) + ceiling(n/2).
If n=0 then a(n) is zero, else a(n) = a(n-1) + v[n mod 2], where v is [3,8]. (End)
G.f.: x*(8 + 3*x) / ((1 + x)*(1 - x)^2). [Bruno Berselli, Jun 16 2014]
a(n) = sum( A010706(i), i=0..n ) - 3. [Bruno Berselli, Jun 16 2014]
E.g.f.: (11*x*exp(x) + 5*sinh(x))/2. - David Lovler, Sep 04 2022

A363861 Sequence related to chains in type D noncrossing partitions.

Original entry on oeis.org

4, 6, 64, 100, 1296, 2058, 32768, 52488, 1000000, 1610510, 35831808, 57921708, 1475789056, 2392031250, 68719476736, 111612119056, 3570467226624, 5808378560022, 204800000000000, 333597619564020, 12855002631049216, 20961814674106394, 876488338465357824, 1430511474609375000, 64509974703297150976

Offset: 3

F. Chapoton, Jun 25 2023

This is counting chains in the noncrossing partition lattices of type D_n that proceed by steps of type A2, except at most one step of type A1 at the end. This is a decomposition number in the terminology of Krattenthaler and Müller.

C. Krattenthaler and T. W. Müller, Decomposition Numbers For Finite Coxeter Groups And Generalised Non-Crossing Partitions, TAMS, vol. 362, 2010.

This is for Coxeter type D what A078707 is for Coxeter type B and A152291 is for Coxeter type A.

print([(n-2)*(n-1)**(n/2-1) if not n % 2 else (n-1)**((n+1)/2) for n in range(3,28)])

a(n) = (n-2)*(n-1)^(n/2-1) if n is even else a(n) = (n-1)^((n+1)/2).

cp's OEIS Frontend

A275549 Number of classes of endofunctions of [n] under reversal.

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A152291 a(n) = (n+1)^floor((n-1)/2).

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Magma

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A092503 a(n) = n^floor(n/2).

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A243520 Numbers that are congruent to {0, 8} mod 11.

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Magma

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A363861 Sequence related to chains in type D noncrossing partitions.

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