A007830 -id:A007830 - cp's OEIS frontend

A008786 a(n) = (n+5)^n.

1, 6, 49, 512, 6561, 100000, 1771561, 35831808, 815730721, 20661046784, 576650390625, 17592186044416, 582622237229761, 20822964865671168, 799006685782884121, 32768000000000000000, 1430568690241985328321, 66249952919459433152512, 3244150909895248285300369

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000169, A000272, A000312, A007778, A007830, A008785, this sequence, A008787, A008788, A008789, A008790, A008791.

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

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

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

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

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

E.g.f.(x) for b(n) = n^(n-5) = a(n-5): T - (15/16)*T^2 + (85/216)T^3 - (25/288)*T^4 + (1/120)*T^5, where T=T(x) is Euler's tree function. - Len Smiley, Nov 17 2001
E.g.f.: LambertW(-x)^5/((-x)^5*(1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
E.g.f.: (1/4)*d/dx((LambertW(-x)/(-x))^4). - Wolfdieter Lang, Oct 25 2022

A008787 a(n) = (n + 6)^n.

Original entry on oeis.org

1, 7, 64, 729, 10000, 161051, 2985984, 62748517, 1475789056, 38443359375, 1099511627776, 34271896307633, 1156831381426176, 42052983462257059, 1638400000000000000, 68122318582951682301, 3011361496339065143296

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, this sequence, A008788, A008789, A008790, A008791.

List([0..20], n-> (n+6)^n); # G. C. Greubel, Sep 11 2019

[(n+6)^n: n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Maple
```
a:= n-> (n+6)^n: seq(a(n), n=0..20);
```

Table[(n+6)^n,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n+5)^(n-1)) \\ G. C. Greubel, Sep 11 2019

[(n+6)^n for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x) for b(n) = n^(n-6) = a(n-6): T - (31/32)*T^2 + (575/1296)*T^3 - (415/3456)*T^4 + (137/7200)*T^5 - (1/720)*T^6; where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
E.g.f.: LambertW(-x)^6/(x^6*(1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
E.g.f.: (1/5)*d/dx(LambertW(-x)/(-x))^5. - Wolfdieter Lang, Oct 25 2022

A008790 a(n) = n^(n+4).

Original entry on oeis.org

0, 1, 64, 2187, 65536, 1953125, 60466176, 1977326743, 68719476736, 2541865828329, 100000000000000, 4177248169415651, 184884258895036416, 8650415919381337933, 426878854210636742656, 22168378200531005859375

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, A008791.

List([0..20], n-> n^(n+4)); # G. C. Greubel, Sep 11 2019

[n^(n+4): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

a:=n->mul(n,k=-3..n):seq(a(n),n=0..20); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+4),{n,0,20}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+3)) \\ G. C. Greubel, Sep 11 2019

[n^(n+4) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.: T*(1 +22*T +58*T^2 +24*T^3)*(1-T)^(-9); where T 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^4/dx^4 {x^4/(T(x)^4*(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

A350452 Number T(n,k) of endofunctions on [n] with exactly k connected components and no fixed points; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 0, 0, 1, 0, 8, 0, 78, 3, 0, 944, 80, 0, 13800, 1810, 15, 0, 237432, 41664, 840, 0, 4708144, 1022252, 34300, 105, 0, 105822432, 27098784, 1286432, 10080, 0, 2660215680, 778128336, 47790540, 648900, 945, 0, 73983185000, 24165049920, 1815578160, 36048320, 138600

Offset: 0

Alois P. Heinz, Dec 31 2021

For k >= 2 and p prime, T(p,k) == 0 (mod 4*p*(p-1)). - Mélika Tebni, Jan 20 2023

			Triangle T(n,k) begins:
  1;
  0;
  0,          1;
  0,          8;
  0,         78,         3;
  0,        944,        80;
  0,      13800,      1810,       15;
  0,     237432,     41664,      840;
  0,    4708144,   1022252,    34300,    105;
  0,  105822432,  27098784,  1286432,  10080;
  0, 2660215680, 778128336, 47790540, 648900, 945;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-1 give: A000007, A000435.
Row sums give A065440.
T(2n,n) gives A001147.
Cf. A060281, A061356, A350446.
Cf. A071720, A007830, A106828.

c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*binomial(n-1, i-1)*x*c(i), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);

c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
     b[n - i]*Binomial[n - 1, i - 1]*x*c[i], {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n/2}]][b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

\\ here AS1(n,k) gives associated Stirling numbers of 1st kind.
AS1(n,k)={(-1)^(n+k)*sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) )}
T(n,k) = {if(n==0, k==0, sum(j=k, n, n^(n-j)*binomial(n-1, j-1)*AS1(j,k)))} \\ Andrew Howroyd, Jan 20 2023

From Mélika Tebni, Jan 20 2023: (Start)
E.g.f. column k: (LambertW(-x) - log(1 + LambertW(-x)))^k / k!.
-Sum_{k=1..n/2} (-1)^k*T(n,k) = A071720(n+1), for n > 0.
-Sum_{k=1..n/2} (-1)^k*T(n,k) / (n-1) = A007830(n-2), for n > 1.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1, j-1)*A106828(j, k) for n > 0. (End)

A060348 a(n) = n^n * (n^2 - 1)/24.

Original entry on oeis.org

9, 160, 3125, 68040, 1647086, 44040192, 1291401630, 41250000000, 1426558353055, 53125098504192, 2120125746145771, 90285055457658880, 4087009643554687500, 195996655783163985920, 9926883142636041170124, 529537075346699234869248

Offset: 3

Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001

nonn

For n >= 3, a(n) is the number of nonequivalent primitive meromorphic functions with one pole of order n on a Riemann surface of genus 1.

B. Shapiro, M. Shapiro and A. Vainshtein, Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs, Amer. Math. Soc. Transl., Vol. 180 (1997), pp. 219-227.

Harry J. Smith, Table of n, a(n) for n = 3..200

Cf. A007830, A060349.

{ for (n=3, 200, write("b060348.txt", n, " ", n^n * (n^2 - 1)/24); ) } \\ Harry J. Smith, Jul 04 2009

A060349 a(n) = n^(n+2)(n^2 - 1)(n+3)(n+2)(5*n - 7)/5760.

Original entry on oeis.org

81, 5824, 328125, 16901136, 847425747, 42630905856, 2186213819427, 115293750000000, 6283133610195442, 354769407810994176, 20781472563720847342, 1263485180096661430272, 79727340621643066406250, 5219469342167970210643968, 354305349685394263423480746

Offset: 3

Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001

nonn

For n >= 3, a(n) is the number of nonequivalent primitive meromorphic functions with one pole of order n on a Riemann surface of genus 2.

B. Shapiro, M. Shapiro and A. Vainshtein, Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs, Amer. Math. Soc. Transl., Vol. 180 (1997), pp. 219-227.

Harry J. Smith, Table of n, a(n) for n = 3..200

Cf. A007830, A060348.

Table[(n^(n+2) (n^2-1)(n+3)(n+2)(5n-7))/5760,{n,3,20}] (* Harvey P. Dale, Jan 10 2013 *)

{ for (n=3, 200, write("b060349.txt", n, " ", n^(n + 2)*(n^2 - 1)*(n + 3)*(n + 2)*(5*n - 7)/5760); ) } \\ Harry J. Smith, Jul 04 2009

A140647 Number of car parking assignments of n cars in n spaces, if one car does not park.

Original entry on oeis.org

1, 10, 107, 1346, 19917, 341986, 6713975, 148717762, 3674435393, 100284451730, 2998366140915, 97510068994690, 3428106725444117, 129590070259759042, 5242767731544271343, 226057515078414496898, 10350122253780835777545, 501543984793431059579122

Offset: 2

Jonathan Vos Post, Jul 09 2008

nonn

			a(3) = 10: [1,3,3], [2,2,2], [2,2,3], [2,3,2], [2,3,3], [3,1,3], [3,2,2], [3,2,3], [3,3,1], [3,3,2].

Alois P. Heinz, Table of n, a(n) for n = 2..380
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg and Pascal Schweitzer, Counting Defective Parking Functions

Cf. A000272.

Column k=1 of A264902.

a(n) = 2*(n+2)^(n-1)-(3*n+1)*(n+1)^(n-2). - Vladeta Jovovic, Jul 21 2008
a(n) = 2*A007830(n-1)-A016777(n)*A007830(n-2). - R. J. Mathar, Aug 09 2008
a(n) ~ (2*exp(2) - 3*exp(1)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017

Extended beyond a(10) by R. J. Mathar, Aug 09 2008

A157994 Number of trees with n edges equipped with a cyclic order on their edges, i.e., number of orbits of the action of Z/nZ on the set of edge-labeled trees of size n, given by cyclically permuting the labels.

Original entry on oeis.org

1, 1, 2, 8, 44, 411, 4682, 66524, 1111134, 21437357, 469070942, 11488238992, 311505013052, 9267596377239, 300239975166840, 10523614185609344, 396861212733968144, 16024522976922760209, 689852631578947368422

Offset: 1

Nikos Apostolakis, Mar 10 2009

A007830, A000169

[1,1] + [((n+1)^(n-2) + sum([(n+1)^(gcd(n,k) -1) for k in [1..n-1]]))/n for  n in [3..20]]

a(1) = 1, a(2) = 1, a(n) = (1/n)*((n+1)^{n-2} + sum_{k=1}^{n-1} (n+1)^{gcd(n,k)-1}) for n > 2

Corrected the formula and Sage code - Nikos Apostolakis, Feb 27 2011.

A173249 Partial sums of A000272.

Original entry on oeis.org

1, 2, 3, 6, 22, 147, 1443, 18250, 280394, 5063363, 105063363, 2463011054, 64380375278, 1856540769315, 58550453144611, 2004745521503986, 74062339559431922, 2936485391069247715, 124376016487663499491

Offset: 0

Jonathan Vos Post, Feb 13 2010

Partial sums of number of trees on n labeled nodes. The subsequence of primes in this sequence begin: 2, 58550453144611, no more through a(30).

			a(19) = 1 + 1 + 1 + 3 + 16 + 125 + 1296 + 16807 + 262144 + 4782969 + 100000000 + 2357947691 + 61917364224 + 1792160394037 + 56693912375296 + 1946195068359375 + 72057594037927936 + 2862423051509815793 + 121439531096594251776 + 5480386857784802185939.

Cf. A000055, A000169, A000312, A007778, A007830, A008785-A008791, A033842, A000272, A036361, A036362, A036506, A000055, A054581, A097170

a(n) = SUM[i=0..n] A000272(i) = SUM[i=0..n] i^(i-2).

A364870 Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 27, 9, 3, 1, 256, 64, 16, 4, 1, 3125, 625, 125, 25, 5, 1, 46656, 7776, 1296, 216, 36, 6, 1, 823543, 117649, 16807, 2401, 343, 49, 7, 1, 16777216, 2097152, 262144, 32768, 4096, 512, 64, 8, 1, 387420489, 43046721, 4782969, 531441, 59049, 6561, 729, 81, 9, 1

Offset: 0

Stefano Spezia, Aug 11 2023

			The array begins:
     1,    1,     1,     1,     1,      1, ...
     1,    2,     3,     4,     5,      6, ...
     4,    9,    16,    25,    36,     49, ...
    27,   64,   125,   216,   343,    512, ...
   256,  625,  1296,  2401,  4096,   6561, ...
  3125, 7776, 16807, 32768, 59049, 100000, ...
  ...

Cf. A000012 (n=0), A000169, A000272, A000312 (k=0), A007830 (k=3), A008785 (k=4), A008786 (k=5), A008787 (k=6), A031973 (antidiagonal sums), A052746 (2nd superdiagonal), A052750, A062971 (main diagonal), A079901 (read by descending antidiagonals), A085527 (1st superdiagonal), A085528 (1st subdiagonal), A085532, A099753.

A[n_,k_]:=(n+k)^n; Join[{1},Table[A[n-k,k],{n,9},{k,0,n}]]//Flatten

E.g.f. of k-th column: LambertW(-x)^k/(x^k*(1 + LambertW(-x))).

cp's OEIS Frontend

A008786 a(n) = (n+5)^n.

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A008787 a(n) = (n + 6)^n.

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A008790 a(n) = n^(n+4).

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A350452 Number T(n,k) of endofunctions on [n] with exactly k connected components and no fixed points; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

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A060348 a(n) = n^n * (n^2 - 1)/24.

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A060349 a(n) = n^(n+2)*(n^2 - 1)*(n+3)*(n+2)*(5*n - 7)/5760.

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A140647 Number of car parking assignments of n cars in n spaces, if one car does not park.

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A060349 a(n) = n^(n+2)(n^2 - 1)(n+3)(n+2)(5*n - 7)/5760.