A000312 -id:A000312 - cp's OEIS frontend

A007781 a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

1, 3, 23, 229, 2869, 43531, 776887, 15953673, 370643273, 9612579511, 275311670611, 8630788777645, 293959006143997, 10809131718965763, 426781883555301359, 18008850183328692241, 808793517812627212561

Offset: 0

Peter McCormack (peter.mccormack(AT)its.csiro.au)

(12n^2 + 6n + 1)^2 divides a(6n+1), where (12n^2 + 6n + 1) = (2n+1)^3 - (2n)^3 = A127854(n) = A003215(2n) are the hex (or centered hexagonal) numbers. The prime numbers of the form 12n^2 + 6n + 1 belong to A002407. - Alexander Adamchuk, Apr 09 2007

			a(14) = 10809131718965763 = 3 * 61^2 * 968299894201.

Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

Doug Bell, Table of n, a(n) for n = 0..100
Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
Ronald K. Hoeflin, Mega Test. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Power Difference Prime.

Cf. A000312, A068146, A068954, A068955, A068956, A068957.

Cf. A002407, A003215, A127854.

[1] cat [(n+1)^(n+1)-n^n: n in [1..20]]; // Vincenzo Librandi, Aug 19 2015

seq( `if`(n=0,1,(n+1)^(n+1) -n^n), n=0..20); # G. C. Greubel, Mar 05 2020

Join[{1},Table[(n+1)^(n+1)-n^n,{n,20}]]  (* Harvey P. Dale, Feb 09 2011 *)
Differences[Table[n^n,{n,0,20}]] (* Charles R Greathouse IV, Feb 09 2011 *)

first(m)=vector(m,i,i--;(i+1)^(i+1) - i^i) /* Anders Hellström, Aug 18 2015 */

[1]+[(n+1)^(n+1) -n^n for n in (1..20)] # G. C. Greubel, Mar 05 2020

a(n) = A000312(n+1) - A000312(n) for n>0, a(0) = 1.
a(n) = abs(discriminant(x^(n+1)-x+1)).
E.g.f.: W(-x)/(1+W(-x)) - W(-x)/((1+W(-x))^3*x) where W is the Lambert W function. - Robert Israel, Aug 19 2015
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

A066068 a(n) = n^n + n.

Original entry on oeis.org

1, 2, 6, 30, 260, 3130, 46662, 823550, 16777224, 387420498, 10000000010, 285311670622, 8916100448268, 302875106592266, 11112006825558030, 437893890380859390, 18446744073709551632, 827240261886336764194

Offset: 0

George E. Antoniou, Dec 02 2001

Harry J. Smith, Table of n, a(n) for n=0..100

Cf. A000169, A000312, A124923, A286193.

[ n^n+n: n in [0..100] ]; // Vincenzo Librandi, Apr 15 2011

Table[ n^n + n, {n, 1, 18} ]
Table[If[n==0,1,n^n+n],{n,0,18}] (* Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *)

a(n) = n^n + n; \\ Harry J. Smith, Nov 09 2009

E.g.f.: (1-x*e^x*T(x)+x*e^x)/(1-T(x)), where T(x) is Euler's tree function (see A000169). - Len Smiley, Dec 04 2001
Resultant of nx^n+1 and nx-1. - Ralf Stephan, Nov 20 2004
a(n) = n*A124923(n), n>0. - R. J. Mathar, Oct 31 2015
Sum_{n>=1} 1/a(n) = A286193. - Amiram Eldar, Nov 19 2020

A241981 Number T(n,k) of endofunctions on [n] where the largest 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, 16, 9, 2, 0, 125, 93, 32, 6, 0, 1296, 1155, 500, 150, 24, 0, 16807, 17025, 8600, 3240, 864, 120, 0, 262144, 292383, 165690, 72030, 24696, 5880, 720, 0, 4782969, 5752131, 3568768, 1719060, 688128, 215040, 46080, 5040

Offset: 0

Alois P. Heinz, Aug 10 2014

T(0,0) = 1 by convention.

In general, number of endofunctions on [n] where the largest cycle length equals k is asymptotic to (k*exp(H(k)) - (k-1)*exp(H(k-1))) * n^(n-1), where H(k) is the harmonic number A001008/A002805, k>=1. The multiplicative constant is (for big k) asymptotic to 2*k*exp(gamma), where gamma is the Euler-Mascheroni constant (see A001620 and A073004). - Vaclav Kotesovec, Aug 21 2014

			Triangle T(n,k) begins:
  1;
  0,      1;
  0,      3,      1;
  0,     16,      9,      2;
  0,    125,     93,     32,     6;
  0,   1296,   1155,    500,   150,    24;
  0,  16807,  17025,   8600,  3240,   864,  120;
  0, 262144, 292383, 165690, 72030, 24696, 5880, 720;
  ...

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

Columns k=0-10 give: A000007, A000272(n+1) for n>0, A163951, A246213, A246214, A246215, A246216, A246217, A246218, A246219, A246220.
T(2n,n) gives A241982.
Row sums give A000312.
Main diagonal gives A000142(n-1) for n>0.
Cf. A246049, A243098.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 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, min(j, k)), j=0..n):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 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, Min[j, k]], {j, 0, n}]; T[0, 0] = 1; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 14, 12, 0, 1, 181, 68, 6, 0, 1, 2584, 520, 20, 0, 0, 1, 41973, 4542, 120, 20, 0, 0, 1, 776250, 46550, 672, 70, 0, 0, 0, 1, 16231381, 540136, 5516, 112, 70, 0, 0, 0, 1, 380333228, 7045020, 40140, 1848, 252, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Jul 30 2014

T(0,0) = 1 by convention.

			T(2,0) = 1: (2,2).
T(2,1) = 2: (1,2), (2,1).
T(2,2) = 1: (1,1).
T(3,1) = 12: (1,1,2), (1,2,1), (1,2,2), (1,2,3), (1,3,2), (2,1,1), (2,1,2), (2,1,3), (2,2,1), (2,3,1), (3,1,2), (3,2,1).
T(3,3) = 1: (1,1,1).
Triangle T(n,k) begins:
0 :         1;
1 :         0,      1;
2 :         1,      2,    1;
3 :        14,     12,    0,   1;
4 :       181,     68,    6,   0,  1;
5 :      2584,    520,   20,   0,  0, 1;
6 :     41973,   4542,  120,  20,  0, 0, 1;
7 :    776250,  46550,  672,  70,  0, 0, 0, 1;
8 :  16231381, 540136, 5516, 112, 70, 0, 0, 0, 1;
     ...

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

Columns k=0-10 give: A133286 (for n>0), A245854, A245855, A245856, A245857, A245858, A245859, A245860, A245861, A245862, A245863.
Row sums give A000312.
T(2n,n) gives A000984(n).
Cf. A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
g:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
T:= (n, k)-> g(n, k) -g(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; g[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[n_, k_] := g[n, k] - g[n, k+1]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

E.g.f. of column k=0: 1 +1/(1+LambertW(-x)) -1/(2-exp(x)); e.g.f. of column k>0: 1/(1-Sum_{j>=k} x^j/j!) - 1/(1-Sum_{j>=k+1} x^j/j!).

T(n,k) = A245732(n,k) - A245732(n,k+1).

A275551 Number of classes of endofunctions of [n] under vertical translation mod n and reversal.

Original entry on oeis.org

1, 1, 2, 6, 36, 325, 3924, 58996, 1049088, 21526641, 500010000, 12968792826, 371504434176, 11649044974645, 396857394156608, 14596463098125000, 576460752571858944, 24330595941321312961, 1092955779880368226560, 52063675149116964615310, 2621440000000512000000000

Offset: 0

Olivier Gérard, Aug 02 2016

nonn

There are two size of classes, n or 2n.
n   c:n    c:2n   (c:n)/n  (c:2n)/n
 1
 1
 2
 3      3      1        1
 8      28     2        7
 25     300    5        60
 72     3852   12       642
 343    58653  49       8379

			a(2) = 2: 11, 12.
a(3) = 6: 111, 112, 113, 121, 123, 131.
a(4) = 36: 1111, 1112, 1113, 1114, 1121, 1122, 1123, 1124, 1131, 1132, 1133, 1134, 1141, 1142, 1143, 1212, 1213, 1214, 1221, 1223, 1224, 1231, 1234, 1241, 1242, 1243, 1312, 1313, 1323, 1324, 1331, 1334, 1341, 1412, 1423, 1441.

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. 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

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

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

A275552 Number of classes of endofunctions of [n] under vertical translation mod n and complement to n+1.

Original entry on oeis.org

1, 1, 2, 5, 36, 313, 3904, 58825, 1048640, 21523361, 500000256, 12968712301, 371504186368, 11649042561241, 396857386631168, 14596463012695313, 576460752303439872, 24330595937833434241, 1092955779869348331520, 52063675148955620766421, 2621440000000000000262144

Offset: 0

Olivier Gérard, Aug 02 2016

There are two size of classes, n or 2n.
.
n   c:n  c:2n   (c:2n)/4
0   1
1   1
2   2
3   1    4      1
4   8    28     7
5   1    312    78
6   32   3872   968
7   1    58824  14706
For n odd, only the set of n constant functions can have a member of their class equal to their complement, so c:n size is 1.
For n even, the c:n class is populated by binary words using k for 0 and n+1-k for 1. There are (2^n)/2 such words as the complement operation identifies them by pairs.
For n odd, c:2n(n) = (n^n - 1*n)/(2*n)
For n even, c:2n(n) = (n^n - 2^(n-1)*n)/(2*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. 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.

a[0] = 1; a[n_?OddQ] := 1 + (n^n - n)/(2n); a[n_?EvenQ] := 2^(n-1) + (n^n - 2^(n-1)*n)/(2n); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 07 2017, translated from PARI *)

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

a(n) = 1 + (n^n - 1*n)/(2*n) if n is odd,

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

A275553 Number of classes of endofunctions of [n] under vertical translation mod n, complement to n+1 and reversal.

Original entry on oeis.org

1, 1, 2, 4, 24, 169, 2024, 29584, 525600, 10764961, 250030128, 6484436676, 185752964096, 5824523694025, 198428723433728, 7298231591777344, 288230377359679488, 12165297972404595841, 546477889989773968640, 26031837574639154232100, 1310720000002816000131072

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

There are three size of classes : n, 2n, 4n.
n   c:n   c:2n   c:4n
----------------------------------
0   1
1   1
2   2
3   1     2      1
4   4     10     10
5   1     24     144
6   8     148    1868
7   1     342    29241
For n odd, only the set of n constant functions can have a member of their class equal to their complement, so c:n size is 1.
For n even, we have 2^(n/2) binary words which have mirror-symmetry
There are three types of classes of size of 2n (stable by reversal, stable by complement, stable by rc as in A275550).

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. 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

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

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

A275554 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and complement to n+1.

Original entry on oeis.org

1, 1, 2, 3, 14, 65, 680, 8407, 131416, 2391515, 50006040, 1178973851, 30958827996, 896080197025, 28346960490560, 973097534189967, 36028797169965112, 1431211525754907905, 60719765554419645244, 2740193428892401092979, 131072000000281600209176

Offset: 0

Olivier Gérard, Aug 02 2016

nonn

Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.
.
n  possible class sizes
-------------------------------
1  1
2  2
3  3, 6,                   18
4  4, 8,      16,          32
5  5, 10,                  50
6  6, 12, 18, 24,     36,  72
7  7, 14,                  98
.
but classes of size 2*n^2 account for the bulk of a(n).
n  number of classes
-----------------------------------
1  1
2  2
3  1, 1,                   1
4  2, 3,       4,          5
5  1, 2,                   62
6  2, 4,   2,  2,     48,  622
7  1, 3,                   8403

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. 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

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

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

A275555 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

Original entry on oeis.org

1, 1, 2, 4, 16, 77, 730, 8578, 132422, 2394795, 50031012, 1179054376, 30959574248, 896082610429, 28346986843640, 973097619619654, 36028798243701780, 1431211529242786625, 60719765604009463866, 2740193429053744941868, 131072000002841600036024

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.
n  possible class sizes
-----------------------------------
1
2
3, 6,   9
4, 8,      16,               32
5, 10,         25,           50
6, 12, 18, 24,     36,       72
7, 14,                 49,   98
but classes of size 2*n^2 account for the bulk of a(n).
n  number of classes
-----------------------------------
1
2
1, 1,   2
2, 3,       8,               3
1, 2,          24,           50
2, 4,  10,  2,    136,       576
1, 3,                 342,   8232

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. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal

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

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

A275556 Number of classes of endofunctions of [n] under vertical translation mod n, rotation, complement to n+1 and reversal.

Original entry on oeis.org

1, 1, 2, 3, 13, 45, 412, 4375, 66988, 1199038, 25033020, 589567451, 15480284910, 448042511917, 14173510363424, 486548852524671, 18014399792942108, 715605766365332673, 30359882832309625502, 1370096714607544395379, 65536000002956800104588

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Because of the interaction between the two symmetries indexed by n and the two involutions, classes can be of size from n up to 4*n^2.
.
n  possible class sizes
------------------------------------
1  1
2  2
3  3, 6,                   18
4  4, 8,      16,          32,   64
5  5, 10,                  50,  100
6  6, 12, 18, 24,     36,  72,  144
7  7, 14,                  98,  196
.
but classes of size 4*n^2 account for the bulk of a(n).
n  number of classes
------------------------------------
1  1
2  2
3  1, 1,                    1
4  2, 3,       4,           3,    1
5  1, 2,                   22,   20
6  2, 4,   2,  2,     28, 116,  258
7  1, 3,                  339, 4032

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. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal

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

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

cp's OEIS Frontend

A007781 a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

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A066068 a(n) = n^n + n.

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A241981 Number T(n,k) of endofunctions on [n] where the largest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

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A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

Mathematica

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A275551 Number of classes of endofunctions of [n] under vertical translation mod n and reversal.

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PARI

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A275552 Number of classes of endofunctions of [n] under vertical translation mod n and complement to n+1.

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Mathematica

PARI

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A275553 Number of classes of endofunctions of [n] under vertical translation mod n, complement to n+1 and reversal.

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