A000312 -id:A000312 - cp's OEIS frontend

A275557 Number of classes of endofunctions of [n] under rotation and complement to n+1.

1, 1, 2, 6, 38, 315, 3932, 58828, 1049108, 21523445, 500010024, 12968712306, 371504436220, 11649042561247, 396857394156656, 14596463012746392, 576460752571867208, 24330595937833434249, 1092955779880370116836, 52063675148955620766430, 2621440000000512000336088

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Classes can be of size 1,2,4, n and 2n.
n   1  2  4      n     2n
--------------------------
1   1
2   0  2
3   1  1                4
4   0  4  4      2     28
5   1  2  0      0    312
6   0  6  6     70   3850
7   1  3  0      0  58824
For n odd, the constant function (n+1)/2 is the only stable by rotation and complement. 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. A275558 Classes under rotation, complement and reversal

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

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

A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 18, 6, 0, 4, 84, 144, 24, 0, 5, 300, 1500, 1200, 120, 0, 6, 930, 10800, 23400, 10800, 720, 0, 7, 2646, 63210, 294000, 352800, 105840, 5040, 0, 8, 7112, 324576, 2857680, 7056000, 5362560, 1128960, 40320

Offset: 0

Philippe Deléham, Dec 14 2003

Another version is in A101817. - Philippe Deléham, Feb 16 2013

			Triangle begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,  18,   6;
  0,  4,  84, 144, 24;
  ...

Alois P. Heinz, Rows n = 0..62, flattened
C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv preprint arXiv:1502.06553, 2015

Row sums give: A000312. Columns k=0-2 give: A000007, A001477, A068605. Diagonal, lower diagonal give: A000142, A001804. Cf. A007318, A048993, A019538, A008279.

Cf. A101817, A288312.

T:= proc(n,k) option remember;
      if k=n then n!
    elif k=0 or k>n then 0
    else n * (T(n-1,k-1) + k/(n-k) * T(n-1,k))
      fi
    end:
seq(seq(T(n,k), k=0..n), n=0..10);

Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 0, n}], {n, 0,10}] // Flatten  (* Geoffrey Critzer, Sep 09 2011 *)

T(n,k) = C(n,k) * k! * A048993(n,k).
T(n,k) = A008279(n,k) * A048993(n,k).
T(n,k) = C(n,k) * A019538(n, k).
T(n,k) = C(n,k) * Sum_{j=0..k} (-1)^(k-j) * C(k,j) * j^n.
T(n,k) = n * (T(n-1,k-1) + k/(n-k) * T(n-1,k)) with T(n,n) = n! and T(n,0) = 0 for n>0.
T(2n,n) = A288312(n). - Alois P. Heinz, Jun 07 2017

Revised description from Jan Maciak, Apr 25 2004

Edited by Alois P. Heinz, Jan 17 2011

A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 10, 1, 1, 6, 27, 35, 1, 1, 10, 93, 256, 126, 1, 1, 18, 381, 2716, 3125, 462, 1, 1, 34, 1785, 36628, 127905, 46656, 1716, 1, 1, 66, 9237, 591460, 7120505, 8848236, 823543, 6435, 1, 1, 130, 51033, 11007556, 495872505, 2443835736, 844691407, 16777216, 24310

Offset: 0

Alois P. Heinz, Jul 21 2014

			A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93.
Square array A(n,k) begins:
0 :    1,    1,      1,       1,         1,           1, ...
1 :    1,    1,      1,       1,         1,           1, ...
2 :    3,    4,      6,      10,        18,          34, ...
3 :   10,   27,     93,     381,      1785,        9237, ...
4 :   35,  256,   2716,   36628,    591460,    11007556, ...
5 :  126, 3125, 127905, 7120505, 495872505, 41262262505, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened

Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245.
Rows n=0+1, 2 give: A000012, A052548.
Main diagonal gives A245398.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.

A245501 Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 27, 1, 1, 1, 4, 10, 256, 1, 1, 1, 3, 19, 41, 3125, 1, 1, 1, 4, 12, 110, 196, 46656, 1, 1, 1, 3, 19, 73, 751, 1057, 823543, 1, 1, 1, 4, 10, 116, 556, 5902, 6322, 16777216, 1, 1, 1, 3, 21, 41, 901, 4737, 52165, 41393, 387420489, 1

Offset: 0

Alois P. Heinz, Jul 24 2014

			Square array A(n,k) begins:
  1,     1,    1,    1,    1,    1,    1, ...
  1,     1,    1,    1,    1,    1,    1, ...
  1,     4,    3,    4,    3,    4,    3, ...
  1,    27,   10,   19,   12,   19,   10, ...
  1,   256,   41,  110,   73,  116,   41, ...
  1,  3125,  196,  751,  556,  901,  220, ...
  1, 46656, 1057, 5902, 4737, 8422, 1921, ...

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

Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506.

Main diagonal gives A245507.

with(numtheory):
A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(
    exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[0, 1] = 1; A[n_, k_] := If[k==0, 1, If[k==1, n^n, n!*SeriesCoefficient[ Exp[ DivisorSum[k-1, (x*Exp[x])^#/#&]], {x, 0, n}]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)

A(n,k) = n! * [x^n] exp(Sum_{d|(k-1)} (x*exp(x))^d/d) for k>1, A(n,0)=1, A(n,1)=n^n.

A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

Original entry on oeis.org

1, 1, 5, 40, 447, 6421, 112726, 2338799, 55990213, 1519122598, 46066158817, 1543974969769, 56677405835276, 2261488166321697, 97455090037460785, 4510770674565054000, 223183550978156866507, 11755122645815049275521, 656670295411196201190366, 38779502115371642484125915, 2413908564514961126280655257

Offset: 0

Ilya Gutkovskiy, Feb 08 2017

Stirling transform of A000312.

			E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 40*x^3/3! + 447*x^4/4! + 6421*x^5/5! + 112726*x^6/6! + ...

G. C. Greubel, Table of n, a(n) for n = 0..375
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's MathWorld, Stirling Transform

Cf. A000312, A000670, A038052, A048802, A052880, A308490, A308491.

b:= proc(n, m) option remember;
     `if`(n=0, m^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2021

Range[0, 20]! CoefficientList[Series[1/(1 + ProductLog[1 - Exp[x]]), {x, 0, 20}], x]
Join[{1}, Table[Sum[StirlingS2[n, k] k^k, {k, 1, n}], {n, 1, 20}]]

x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(1-exp(x))))) \\ G. C. Greubel, Nov 12 2017

a(0) = 1, a(n) = Sum_{k=1..n} Stirling2(n,k)*k^k.

a(n) ~ n^n / (sqrt(1+exp(1)) * (log(1+exp(-1)))^(n+1/2) * exp(n)). - Vaclav Kotesovec, Feb 17 2017

A008789 a(n) = n^(n+3).

Original entry on oeis.org

0, 1, 32, 729, 16384, 390625, 10077696, 282475249, 8589934592, 282429536481, 10000000000000, 379749833583241, 15407021574586368, 665416609183179841, 30491346729331195904, 1477891880035400390625, 75557863725914323419136

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, A008790, A008791.

List([0..20], n-> n^(n+3)); # G. C. Greubel, Sep 11 2019

[n^(n+3): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

printlevel := -1; a := [0]; T := x->-LambertW(-x); f := series((T(x)*(1+8*T(x)+6*(T(x))^2)/(1-T(x))^7),x,24); for m from 1 to 23 do a := [op(a),op(2*m-1,f)*m! ] od; print(a); # Len Smiley, Nov 19 2001

Table[n^(n+3),{n,0,20}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+2)) \\ G. C. Greubel, Sep 11 2019

[n^(n+3) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x): T*(1 +8*T +6*T^2)*(1-T)^(-7); where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 19 2001
See A008517 and A134991 for similar e.g.f.s and diagonals of A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^3/dx^3 {x^3/(T(x)^3*(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
a(n) = n*A008788(n). - R. J. Mathar, Oct 31 2015

A055137 Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -2, -3, 0, 1, -3, -8, -6, 0, 1, -4, -15, -20, -10, 0, 1, -5, -24, -45, -40, -15, 0, 1, -6, -35, -84, -105, -70, -21, 0, 1, -7, -48, -140, -224, -210, -112, -28, 0, 1, -8, -63, -216, -420, -504, -378, -168, -36, 0, 1, -9, -80, -315, -720

Offset: 0

Christian G. Bower, Apr 25 2000

The n-th row consists of coefficients of the characteristic polynomial of the adjacency matrix of the complete n-graph.
Triangle of coefficients of det(M(n)) where M(n) is the n X n matrix m(i,j)=x if i=j, m(i,j)=i/j otherwise. - Benoit Cloitre, Feb 01 2003
T is an example of the group of matrices outlined in the table in A132382--the associated matrix for rB(0,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) *(1-x). T(n,k) = Binomial(n,k)* s(n-k) where s = (1,0,-1,-2,-3,...) with an e.g.f. of exp(x)*(1-x) which is the reciprocal of the e.g.f. of A000166. - Tom Copeland, Sep 10 2008
Row sums are: {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}. - Roger L. Bagula, Feb 20 2009
T is related to an operational calculus connecting an infinitesimal generator for fractional integro-derivatives with the values of the Riemann zeta function at positive integers (see MathOverflow links). - Tom Copeland, Nov 02 2012
The submatrix below the null subdiagonal is signed and row reversed A127717. The submatrix below the diagonal is A074909(n,k)s(n-k) where s(n)= -n, i.e., multiply the n-th diagonal by -n. A074909 and its reverse A135278 have several combinatorial interpretations. - Tom Copeland, Nov 04 2012
T(n,k) is the difference between the number of even (A145224) and odd (A145225) permutations (of an n-set) with exactly k fixed points. - Julian Hatfield Iacoponi, Aug 08 2024

			1; 0,1; -1,0,1; -2,-3,0,1; -3,-8,-6,0,1; ...
(Bagula's matrix has a different sign convention from the list.)
From _Roger L. Bagula_, Feb 20 2009: (Start)
  { 1},
  { 0,   1},
  {-1,   0,    1},
  { 2,  -3,    0,    1},
  {-3,   8,   -6,    0,     1},
  { 4, -15,   20,  -10,     0,    1},
  {-5,  24,  -45,   40,   -15,    0,    1},
  { 6, -35,   84, -105,    70,  -21,    0,   1},
  {-7,  48, -140,  224,  -210,  112,  -28,   0,   1},
  { 8, -63,  216, -420,   504, -378,  168, -36,   0, 1},
  {-9,  80, -315,  720, -1050, 1008, -630, 240, -45, 0, 1}
(End)
R(3,x) = (-1)^3*Sum_{permutations p in S_3} sign(p)*(-x)^(fix(p)).
    p   | fix(p) | sign(p) | (-1)^3*sign(p)*(-x)^fix(p)
========+========+=========+===========================
  (123) |    3   |    +1   |      x^3
  (132) |    1   |    -1   |       -x
  (213) |    1   |    -1   |       -x
  (231) |    0   |    +1   |       -1
  (312) |    0   |    +1   |       -1
  (321) |    1   |    -1   |       -x
========+========+=========+===========================
                           | R(3,x) = x^3 - 3*x - 2
- _Peter Bala_, Aug 08 2011

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. p. 17.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p.184 problem 3.

Problem B6, The 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005
M. Bhargava, K. Kedlaya, and L. Ng, Solutions to the 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005
T. Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants
T. Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

Cf. A005563, A005564 (absolute values of columns 1, 2).
Cf. A008290, A133314, A238363, A238385.
Cf. A000312.
Cf. A145224, A145225, A003221, A000387.

M[n_] := Table[If[i == j, x, 1], {i, 1, n}, {j, 1, n}]; a = Join[{{1}}, Flatten[Table[CoefficientList[Det[M[n]], x], {n, 1, 10}]]] (* Roger L. Bagula, Feb 20 2009 *)
t[n_, k_] := (k-n+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2013, after Pari *)

T(n,k)=(1-n+k)*if(k<0 || k>n,0,n!/k!/(n-k)!)

G.f.: (x-n+1)*(x+1)^(n-1) = Sum_(k=0..n) T(n,k) x^k.
T(n, k) = (1-n+k)*binomial(n, k).
k-th column has o.g.f. x^k(1-(k+2)x)/(1-x)^(k+2). k-th row gives coefficients of (x-k)(x+1)^k. - Paul Barry, Jan 25 2004
T(n,k) = Coefficientslist[Det[Table[If[i == j, x, 1], {i, 1, n}, {k, 1, n}],x]. - Roger L. Bagula, Feb 20 2009
From Peter Bala, Aug 08 2011: (Start)
Given a permutation p belonging to the symmetric group S_n, let fix(p) be the number of fixed points of p and sign(p) its parity. The row polynomials R(n,x) have a combinatorial interpretation as R(n,x) = (-1)^n*Sum_{permutations p in S_n} sign(p)*(-x)^(fix(p)). An example is given below.
Note: The polynomials P(n,x) = Sum_{permutations p in S_n} x^(fix(p)) are the row polynomials of the rencontres numbers A008290. The integral results Integral_{x = 0..n} R(n,x) dx = n/(n+1) = Integral_{x = 0..-1} R(n,x) dx lead to the identities Sum_{p in S_n} sign(p)*(-n)^(1 + fix(p))/(1 + fix(p)) = (-1)^(n+1)*n/(n+1) = Sum_{p in S_n} sign(p)/(1 + fix(p)). The latter equality was Problem B6 in the 66th William Lowell Putnam Mathematical Competition 2005. (End)
From Tom Copeland, Jul 26 2017: (Start)
The e.g.f. in Copeland's 2008 comment implies this entry is an Appell sequence of polynomials P(n,x) with lowering and raising operators L = d/dx and R = x + d/dL log[exp(L)(1-L)] = x+1 - 1/(1-L) = x - L - L^2 - ... such that L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
P(n,x) = (1-L) exp(L) x^n = (1-L) (x+1)^n = (x+1)^n - n (x+1)^(n-1) = (x+1-n)(x+1)^(n-1) = (x+s.)^n umbrally, where (s.)^n = s_n = P(n,0).
The formalism of A133314 applies to the pair of entries A008290 and A055137.
The polynomials of this pair P_n(x) and Q_n(x) are umbral compositional inverses; i.e., P_n(Q.(x)) = x^n = Q_n(P.(x)), where, e.g., (Q.(x))^n = Q_n(x).
The exponential infinitesimal generator (infinigen) of this entry is the negated infinigen of A008290, the matrix (M) noted by Bala, related to A238363. Then e^M = [the lower triangular A008290], and e^(-M) = [the lower triangular A055137]. For more on the infinigens, see A238385. (End)
From the row g.f.s corresponding to Bagula's matrix example below, the n-th row polynomial has a zero of multiplicity n-1 at x = 1 and a zero at x = -n+1. Since this is an Appell sequence dP_n(x)/dx = n P_{n-1}(x), the critical points of P_n(x) have the same abscissas as the zeros of P_{n-1}(x); therefore, x = 1 is an inflection point for the polynomials of degree > 2 with P_n(1) = 0, and the one local extremum of P_n has the abscissa x = -n + 2 with the value (-n+1)^{n-1}, signed values of A000312. - Tom Copeland, Nov 15 2019
From Julian Hatfield Iacoponi, Aug 08 2024: (Start)
T(n,k) = A145224(n,k) - A145225(n,k).
T(n,k) = binomial(n,k)*(A003221(n-k)-A000387(n-k)). (End)

Additional comments from Michael Somos, Jul 04 2002

A055858 Coefficient triangle for certain polynomials.

Original entry on oeis.org

1, 1, 2, 4, 9, 6, 27, 64, 48, 36, 256, 625, 500, 400, 320, 3125, 7776, 6480, 5400, 4500, 3750, 46656, 117649, 100842, 86436, 74088, 63504, 54432, 823543, 2097152, 1835008, 1605632, 1404928, 1229312, 1075648, 941192, 16777216, 43046721

Offset: 0

Wolfdieter Lang, Jun 20 2000

The coefficients of the partner polynomials are found in triangle A055864.

			{1}; {1,2}; {4,9,6}; {27,64,48,36}; ...
Fourth row polynomial (n=3): p(3,x) = 27 + 64*x + 48*x^2 + 36*x^3.

Column sequences are A000312(n), n >= 1, A055860 (A000169), A055861 (A053506), A055862-3 for m=0..4, row sums: A045531(n+1)= |A039621(n+1, 2)|, n >= 0.

a[n_, m_] /; n < m = 0; a[0, 0] = 1; a[n_, 0] := n^n; a[n_, m_] := n^(m-1)*(n+1)^(n-m+1); Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2013 *)

a(n, m)=0 if n < m; a(0, 0)=1, a(n, 0) = n^n, n >= 1, a(n, m) = n^(m-1)*(n+1)^(n-m+1), n >= m >= 1;

E.g.f. for column m: A(m, x); A(0, x) = 1/(1+W(-x)); A(1, x) = -1 - (d/dx)W(-x) = -1-W(-x)/((1+W(-x))*x); A(2, x) = A(1, x)-int(A(1, x), x)/x-(1/x+x); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-((m-1)^(m-1))*(x^(m-1))/(m-1)!, m >= 3; W(x) principal branch of Lambert's function.

A062481 a(n) = n^prime(n).

Original entry on oeis.org

1, 8, 243, 16384, 48828125, 13060694016, 232630513987207, 144115188075855872, 8862938119652501095929, 100000000000000000000000000000, 191943424957750480504146841291811, 8505622499821102144576131684114829934592, 4695452425098908797088971409337422035076128813

Offset: 1

Labos Elemer, Jul 09 2001

Harry J. Smith, Table of n, a(n) for n = 1..50

Cf. A000040, A000312, A051674, A096250.

Table[n^Prime[n],{n,20}] (* Harvey P. Dale, Jun 12 2014 *)

a(n)={n^prime(n)} \\ Harry J. Smith, Aug 08 2009

from sympy import prime
def a(n): return n**prime(n)
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Jun 15 2022

From Amiram Eldar, Nov 18 2020: (Start)
a(n) = n^A000040(n).
Sum_{n>=1} 1/a(n) = A096250. (End)
a(n) == n (mod prime(n)). [Fermat's little theorem] - Nicolas Bělohoubek, Jun 14 2022

More terms from Harvey P. Dale, Jun 12 2014

A066588 a(n) = sum of the digits of n^n.

Original entry on oeis.org

1, 1, 4, 9, 13, 11, 27, 25, 37, 45, 1, 41, 54, 58, 52, 99, 88, 98, 108, 127, 31, 117, 148, 146, 153, 151, 154, 189, 163, 167, 63, 184, 205, 207, 214, 260, 270, 271, 265, 306, 112, 308, 315, 313, 325, 306, 352, 374, 333, 355, 151, 414, 412, 350, 378, 442, 391, 450

Offset: 0

Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 07 2002

			a(7) = 25 because 7^7 = 823543 and 8 + 2 + 3 + 5 + 4 + 3 = 25.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000312, A007953, A004152.

[&+Intseq((n^n)): n in [0..80] ]; // Vincenzo Librandi, Jun 18 2015

a:= n-> add(i, i=convert(n^n, base, 10)):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 06 2023

Table[Plus@@IntegerDigits@(n^n), {n, 80}] (* Vincenzo Librandi, Jun 18 2015 *)

a(n) = sumdigits(n^n) \\ Michel Marcus, Jun 18 2015

More terms from Paolo P. Lava, May 15 2007

a(0)=1 inserted by Sean A. Irvine, Oct 06 2023

cp's OEIS Frontend

A275557 Number of classes of endofunctions of [n] under rotation and complement to n+1.

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A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).

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A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A245501 Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

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A008789 a(n) = n^(n+3).

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A055137 Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.

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