A000312 -id:A000312 - cp's OEIS frontend

A275558 Number of classes of endofunctions of [n] under rotation, complement to n+1 and reversal.

1, 1, 2, 6, 31, 195, 2182, 30100, 529674, 10778125, 250155012, 6484839306, 185757443582, 5824538174455, 198428907905336, 7298232189810696, 288230385949610020, 12165298000307625609, 546477890436083284338, 26031837576091248872110, 1310720000028416000168044

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Classes can be of size 1,2,4, n, 2n or 4n.
.
n   1  2  4      n     2n     4n
---------------------------------
1   1
2   0  2
3   1  1                4
4   0  4  4      0     17      6
5   1  2  0      0     72    120
6   0  6  6     30    410   1730
7   1  3  0      0   1368  28728
.
For n odd, the constant function (n+1)/2 is the only stable by rotation, complement and reversal. So #c1=1.
For n even, there is no stable function, so #c1=0, but constant functions are grouped two by two making n/2 classes of size 2. Functions alternating a value and its complement are also grouped two by two, making another n/2 classes. This gives #c2=n.

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A275549 Classes under 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

\\ see A056391 for Polya enumeration functions
a(n) = NonequivalentSorts(DihedralPerms(n), ReversiblePerms(n)); \\ Andrew Howroyd, Sep 30 2017

Terms a(8) and beyond from Andrew Howroyd, Sep 30 2017

A008791 a(n) = n^(n+5).

Original entry on oeis.org

0, 1, 128, 6561, 262144, 9765625, 362797056, 13841287201, 549755813888, 22876792454961, 1000000000000000, 45949729863572161, 2218611106740436992, 112455406951957393129, 5976303958948914397184, 332525673007965087890625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, A008787, A008788, A008789, A008790.

List([0..20], n-> n^(n+5)); # G. C. Greubel, Sep 11 2019

[n^(n+5): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

a:=n->mul( n, k=-4..n): seq(a(n), n=0..20); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+5),{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+4)) \\ G. C. Greubel, Sep 11 2019

[n^(n+5) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x): T*(1 + 52*T + 328*T^2 + 444*T^3 + 120*T^4)*(1-T)^(-11); where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
See A008517 and A134991 for similar e.g.f.s and diagonals of A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^5/dx^5 {x^5/(T(x)^5*(1-T(x)))}, where T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012

A008973 Fibonacci number F(n) to power F(n).

Original entry on oeis.org

1, 1, 1, 4, 27, 3125, 16777216, 302875106592253, 5842587018385982521381124421, 11756638905368616011414050501310355554617941909569536, 524744532468751923546122657597368049278513737089035272057324643668607677682302892208099365234375

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..14

Cf. A000045, A000312, A250495.

with(combinat):
[ seq(fibonacci(n)^fibonacci(n), n=0..15) ];

Table[f=Fibonacci[n];f^f,{n,0,9}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)

a(n)=my(f=fibonacci(n)); f^f \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A000045(n)^A000045(n).

a(n) = A000312(A000045(n)).

A091512 a(n) is the largest integer m such that 2^m divides (2n)^n, i.e., the exponent of 2 in (2n)^n.

Original entry on oeis.org

1, 4, 3, 12, 5, 12, 7, 32, 9, 20, 11, 36, 13, 28, 15, 80, 17, 36, 19, 60, 21, 44, 23, 96, 25, 52, 27, 84, 29, 60, 31, 192, 33, 68, 35, 108, 37, 76, 39, 160, 41, 84, 43, 132, 45, 92, 47, 240, 49, 100, 51, 156, 53, 108, 55, 224, 57, 116, 59, 180, 61, 124, 63

Offset: 1

Ralf Stephan and Labos Elemer, Jan 18 2004

n times one more than the trailing 0's in the binary representation of n. - Ralf Stephan, Aug 22 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n.

Cf. A091519, A090740, A090739, A048298, A162728, A220466.

[n*(Valuation(n, 2)+1): n in [1..80]]; // Vincenzo Librandi, May 16 2013

nmax:=63: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (2*n-1)*(p+1)*2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 08 2013
# second Maple program:
a:= n-> n*padic[ordp](2*n, 2):
seq(a(n), n=1..63);  # Alois P. Heinz, Jan 02 2024

Table[ Part[ Flatten[ FactorInteger[(2 n)^n]], 2], {n, 1, 124}]
Table[IntegerExponent[(2n)^n,2],{n,70}] (* Harvey P. Dale, Sep 11 2015 *)

PARI
```
a(n)=n*(valuation(n,2)+1)
```
PARI
```
a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+n,n))
```

def A091512(n): return n*(n&-n).bit_length() # Chai Wah Wu, Jul 11 2022

a(n) = A007814(A000312(n)) = n*A001511(n) = A069895(n)/2.
G.f.: Sum_{k>=0} 2^k*x^2^k/(1-x^2^k)^2.
Recurrence: a(0) = 0, a(2*n) = 2*a(n) + 2*n, a(2*n+1) = 2*n+1.
Dirichlet g.f.: zeta(s-1)*2^s/(2^s-2). - Ralf Stephan, Jun 17 2007
Mobius transform of A162728, where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
a(n) = A162728(2*n)/phi(2*n), where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
a((2*n-1)*2^p) = (2*n-1)*(p+1)*2^p, p >= 0. Observe that a(2^p) = A001787(p+1). - Johannes W. Meijer, Feb 08 2013
Sum_{k=1..n} a(k) ~ n^2. - Amiram Eldar, Oct 22 2022
a(n) = Sum_{d divides n} d*A048298(n/d); that is, a(n) is the Dirichlet product of A048298(n) and A000027(n). - Peter Bala, Jan 02 2024

A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 24, 0, 3, 0, 240, 12, 0, 4, 0, 3080, 40, 0, 0, 5, 0, 46410, 210, 30, 0, 0, 6, 0, 822612, 840, 84, 0, 0, 0, 7, 0, 16771832, 5208, 112, 56, 0, 0, 0, 8, 0, 387395856, 23760, 720, 144, 0, 0, 0, 0, 9, 0, 9999848700, 148410, 2610, 180, 90, 0, 0, 0, 0, 10

Offset: 0

Alois P. Heinz, Aug 27 2013

Sum_{k=0..n} k*T(n,k) = A228618(n).
Sum_{k=0..n}   T(n,k) = A000312(n).
T(2*n,n)   = A002939(n) for n>0.
T(2*n+1,n) = A033586(n) for n>1.
T(2*n+2,n) = A085250(n+1) for n>2.
T(2*n+3,n) = A033586(n+1) for n>3.

			T(3,1) = 24: [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [1,3,3], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,1], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,    2;
  0,       24,    0,   3;
  0,      240,   12,   0,  4;
  0,     3080,   40,   0,  0,  5;
  0,    46410,  210,  30,  0,  0,  6;
  0,   822612,  840,  84,  0,  0,  0,  7;
  0, 16771832, 5208, 112, 56,  0,  0,  0,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-10 give: A000007, A228619, A228621, A228622, A228630, A228631, A228632, A228633, A228634, A228635, A228636.
Main diagonal gives: A028310.
Cf. A228154, A228273.

A242027 Number T(n,k) of endofunctions on [n] with cycles of k distinct lengths; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 4, 0, 24, 3, 0, 206, 50, 0, 2300, 825, 0, 31742, 14794, 120, 0, 522466, 294987, 6090, 0, 9996478, 6547946, 232792, 0, 218088504, 160994565, 8337420, 0, 5344652492, 4355845868, 299350440, 151200, 0, 145386399554, 128831993037, 11074483860, 18794160

Offset: 0

Alois P. Heinz, Aug 11 2014

			T(3,2) = 3: (1,3,2), (3,2,1), (2,1,3).
Triangle T(n,k) begins:
00 :  1;
01 :  0,          1;
02 :  0,          4;
03 :  0,         24,          3;
04 :  0,        206,         50;
05 :  0,       2300,        825;
06 :  0,      31742,      14794,       120;
07 :  0,     522466,     294987,      6090;
08 :  0,    9996478,    6547946,    232792;
09 :  0,  218088504,  160994565,   8337420;
10 :  0, 5344652492, 4355845868, 299350440, 151200;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A241980 for n>0, A246283, A246284, A246285, A246286, A246287, A246288, A246289, A246290, A246291.
Row sums give A000312.
T(A000217(n),n) gives A246292.
Cf. A003056, A060281, A218868 (the same for permutations).

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i<1 or k<1, 0, add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1, k-`if`(j=0, 0, 1)), j=0..n/i)))
    end:
T:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2, k), j=0..n):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);

multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k==0, 1, 0], If[i<1 || k<1, 0, Sum[(i-1)!^j*multinomial[n, Join[ {n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, k-If[j==0, 0, 1]], {j, 0, n/i}]] ]; T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, j, k], {j, 0, n}]; Table[T[n, k], {n, 0, 14}, {k, 0, Floor[(Sqrt[1+8n]-1)/2]}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

A245405 Number A(n,k) of endofunctions on [n] such that no element has a preimage of cardinality k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 2, 6, 1, 1, 2, 3, 24, 1, 1, 4, 9, 40, 120, 1, 1, 4, 24, 76, 205, 720, 1, 1, 4, 27, 208, 825, 2556, 5040, 1, 1, 4, 27, 252, 2325, 10206, 24409, 40320, 1, 1, 4, 27, 256, 3025, 31956, 143521, 347712, 362880, 1, 1, 4, 27, 256, 3120, 44406, 520723, 2313200, 4794633, 3628800

Offset: 0

Alois P. Heinz, Jul 21 2014

			Square array A(n,k) begins:
0 :   1,    1,     1,     1,     1,     1,     1, ...
1 :   1,    0,     1,     1,     1,     1,     1, ...
2 :   2,    2,     2,     4,     4,     4,     4, ...
3 :   6,    3,     9,    24,    27,    27,    27, ...
4 :  24,   40,    76,   208,   252,   256,   256, ...
5 : 120,  205,   825,  2325,  3025,  3120,  3125, ...
6 : 720, 2556, 10206, 31956, 44406, 46476, 46650, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Column k=0-10 give: A000142, A231797, A245406, A245407, A245408, A245409, A245410, A245411, A245412, A245413, A245414.
Main diagonal gives A061190.
A(n,n+1) gives A000312.

b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1,
      `if`(i<1, 0, add(`if`(j=k, 0, b(n-j, i-1, k)*
       binomial(n, j)), j=0..n)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

nn = n; f[m_]:=Flatten[Table[m[[j, i - j + 1]], {i, 1, Length[m]}, {j, 1, i}]]; f[Transpose[Table[Prepend[Table[n! Coefficient[Series[(Exp[x] -x^k/k!)^n, {x, 0, nn}],x^n], {n, 1, 10}], 1], {k, 0, 10}]]] (* Geoffrey Critzer, Jan 31 2015 *)

A(n,k) = n! * [x^n] (exp(x)-x^k/k!)^n.

A246049 Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 19, 6, 2, 0, 175, 51, 24, 6, 0, 2101, 580, 300, 120, 24, 0, 31031, 8265, 4360, 2160, 720, 120, 0, 543607, 141246, 74130, 41160, 17640, 5040, 720, 0, 11012415, 2810437, 1456224, 861420, 430080, 161280, 40320, 5040

Offset: 0

Alois P. Heinz, Aug 11 2014

T(0,0) = 1 by convention.

In general, number of endofunctions on [n] where the smallest cycle length equals k is asymptotic to (exp(-H(k-1)) - exp(-H(k))) * n^n, where H(k) is the harmonic number A001008/A002805, k>=1. - Vaclav Kotesovec, Aug 21 2014

			Triangle T(n,k) begins:
  1;
  0,      1;
  0,      3,      1;
  0,     19,      6,     2;
  0,    175,     51,    24,     6;
  0,   2101,    580,   300,   120,    24;
  0,  31031,   8265,  4360,  2160,   720,  120;
  0, 543607, 141246, 74130, 41160, 17640, 5040, 720;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A045531, A246189, A246190, A246191, A246192, A246193, A246194, A246195, A246196, A246197.
T(2n,n) gives A246050.
Row sums give A000312.
Main diagonal gives A000142(n-1) for n>0.
Cf. A241981, A243098.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
      add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i+1), j=0..n/i)))
    end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), A(n, k) -A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0,
      Sum[(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!*
      b[n-i*j, i+1], {j, 0, n/i}]]];
A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, k], {j, 0, n}];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], A[n, k] - A[n, k+1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

A275550 Number of classes of endofunctions of [n] under reversal and complement to n+1.

Original entry on oeis.org

1, 1, 2, 10, 72, 819, 11772, 206572, 4196352, 96871525, 2500050000, 71328400806, 2229026605056, 75718793541895, 2778001759096256, 109473473278652344, 4611686020574871552, 206810065502975099529

Offset: 0

Olivier Gérard, Aug 01 2016

Possible classes size are 1,2,4
n 1 2    4
-----------------
1 0    0
0 2    0
1 5    4
0 16   56
1 74   744
0 216  11556
1 1371 205200.
Classes of size 2 can be further decomposed by whether the function is stable by reversal or stable by (reversal and complement).
n 2    2-r  2-rc
-----------------
0    0    0
2    1    1
5    4    1
16   8    8
74   62   12
216  108  108
1371 1200 171.

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A275549 Classes under 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. A192396 floor(((k+1)^n-(1+(-1)^k)/2)/2)
Cf. A275574 (2-r classes)

Table[1/8 (1+(-1)^(1+n)+2 n^n+n^Floor[n/2] (3+(-1)^(n+1) (-1+n)+n)),{n,1,17}]

a(n) = (1+(-1)^(n+1)+2*n^n+(3+((-1)^(n+1))*(n-1)+n)*n^(floor(n/2)) )/8; \\ Andrew Howroyd, Sep 30 2017

a(n) = (1+(-1)^(n+1)+2*n^n+(3+((-1)^(n+1))*(n-1)+n)*n^(floor(n/2)) )/8.

Classes of size 2: (2 (-1 + (-1)^n) + n^floor(n/2)*(3 + ((-1)^(1 + n))* (-1 + n) + n))/4.

Duplicate a(7) removed by Andrew Howroyd, Sep 30 2017

A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 10, 2, 1, 1, 8, 7, 21, 5, 20, 3, 4, 1, 1, 9, 8, 28, 6, 35, 4, 10, 3, 1, 1, 10, 9, 36, 7, 56, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 84, 6, 35, 10, 10, 2, 1

Offset: 1

Gus Wiseman, Apr 29 2021

First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.

As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4  10  20  35  56  84 120 165
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
  n=9:  1   3   6  10  15  21  28  36  45
Triangle begins:
   1
   1   1
   1   2   1
   1   3   2   1
   1   4   3   3   1
   1   5   4   6   2   1
   1   6   5  10   3   4   1
   1   7   6  15   4  10   2   1
   1   8   7  21   5  20   3   4   1
   1   9   8  28   6  35   4  10   3   1
   1  10   9  36   7  56   5  20   6   4   1
   1  11  10  45   8  84   6  35  10  10   2   1
For example, row n = 6 counts the following multisets:
  {1,1,1,1,1}  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
               {1,1,1,2}  {1,1,3}  {1,2}  {5}
               {1,1,2,2}  {1,3,3}  {1,4}
               {1,2,2,2}  {3,3,3}  {2,2}
               {2,2,2,2}           {2,4}
                                   {4,4}
Note that for n = 6, k = 4 in the triangle, the two multisets {1,4} and {2,2} represent the same divisor 4, so they are only counted once under A343656(4,2) = 5.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Row k = 1 of the array is A000005.
Column n = 4 of the array is A000217.
Column n = 6 of the array is A000292.
Row k = 2 of the array is A184389.
The distinct products of these multisets are counted by A343656.
Antidiagonal sums of the array (or row sums of the triangle) are A343661.
A000312 = n^n.
A009998(n,k) = n^k (as an array, offset 1).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,k],n-k],{n,10},{k,n}]

A(n,k) = binomial(numdiv(n) + k - 1, k)
{ for(n=1, 9, for(k=0, 8, print1(A(n,k), ", ")); print ) } \\ Andrew Howroyd, Jan 11 2024

A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).

T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).

cp's OEIS Frontend

A275558 Number of classes of endofunctions of [n] under rotation, complement to n+1 and reversal.

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A008791 a(n) = n^(n+5).

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A008973 Fibonacci number F(n) to power F(n).

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A091512 a(n) is the largest integer m such that 2^m divides (2*n)^n, i.e., the exponent of 2 in (2*n)^n.

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A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A242027 Number T(n,k) of endofunctions on [n] with cycles of k distinct lengths; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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A245405 Number A(n,k) of endofunctions on [n] such that no element has a preimage of cardinality k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A091512 a(n) is the largest integer m such that 2^m divides (2n)^n, i.e., the exponent of 2 in (2n)^n.