A008790 -id:A008790 - cp's OEIS frontend

A007778 a(n) = n^(n+1).

0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632

Offset: 0

N. J. A. Sloane

Number of edges of the complete bipartite graph of order n+n^n, K_n,n^n. - Roberto E. Martinez II, Jan 07 2002
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n >= 1. - Nick Hobson, Nov 30 2006
a(n) is also the number of ways of writing an n-cycle as the product of n+1 transpositions. - Nikos Apostolakis, Nov 22 2008
a(n) is the total number of elements whose preimage is the empty set summed over all partial functions from [n] into [n]. - Geoffrey Critzer, Jan 12 2022

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 67.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Nick Hobson, Exponential equation.
Yidong Sun and Jujuan Zhuang, lambda-factorials of n, arXiv:1007.1339 [math.CO], 2010. - _Peter Luschny_, Jul 09 2010

Cf. A000169, A000272, A000312, A007830, A008785, A008786, A008787, A008788, A008789, A008790, A008791, A135608.
Essentially the same as A065440.
Cf. A061250, A143857. [From Reinhard Zumkeller, Jul 23 2010]

[n^(n+1):n in [0..20]]; // Vincenzo Librandi, Jan 03 2012

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

Table[n^(n+1), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)

A007778[n]:=n^(n+1)$
makelist(A007778[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

vector(21, n, my(m=n-1); m^(m+1)) \\ G. C. Greubel, Mar 05 2020

[n^(n+1) for n in (0..20)] # G. C. Greubel, Mar 05 2020

E.g.f.: -W(-x)/(1 + W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(k+1)*(n+1)^(n-k). - Peter Luschny, Jul 09 2010
See A008517 and A134991 for similar e.g.f.s. and A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d/dx {x/(T(x)*(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*A000312(n). - R. J. Mathar, Jan 12 2017
Sum_{n>=2} 1/a(n) = A135608. - Amiram Eldar, Nov 17 2020

A007830 a(n) = (n+3)^n.

Original entry on oeis.org

1, 4, 25, 216, 2401, 32768, 531441, 10000000, 214358881, 5159780352, 137858491849, 4049565169664, 129746337890625, 4503599627370496, 168377826559400929, 6746640616477458432, 288441413567621167681, 13107200000000000000000, 630880792396715529789561

Offset: 0

Peter J. Cameron, Mar 15 1996

a(n-2) is the number of trees with n+1 unlabeled vertices and n labeled edges for n > 1. - Christian G. Bower, 12/99 [corrected by Jonathan Vos Post, Sep 22 2012]
a(n) is the number of nonequivalent primitive meromorphic functions with one pole of order n+3 on a Riemann surface of genus 0. - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001
Pikhurko writes: "Cameron demonstrated that the total number of edge-labeled trees with n >= 2 edges is (n+1)^(n-2) by showing that the number of vertex-labeled trees of size n is n+1 times larger than the number of edge-labeled ones." - Jonathan Vos Post, Sep 22 2012
With offset 1, a(n) is the number of ways to build a rooted labeled forest with some (possibly all or none) of the nodes from {1,2,...,n} and then build another forest with the remaining nodes. - Geoffrey Critzer, May 10 2013

M. Shapiro, B. 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.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.27.

T. D. Noe, Table of n, a(n) for n = 0..100
Christian Brouder, William J. Keith, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
P. J. Cameron, Two-graphs and Trees, Discrete Math. 127 (1994) 63-74.
P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
Oleg Pikhurko, Generating Edge-Labeled Trees, American Math. Monthly, 112 (2005) 919-921.
M. Shapiro, B. Shapiro and A. Vainshtein, Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs, 1996.
Index entries for sequences related to trees

Cf. A000169, A000272, A000312, A007778.

Cf. A008785, A008786, A008787, A008788, A008789, A008790, A008791.

[(n+3)^n: n in [0..20]]; // G. C. Greubel, Mar 06 2020

A007830:=n->(n+3)^n; seq(A007830(n), n=0..20);
T := -LambertW(-x): ser := series(exp(3*T)/(1-T), x, 20):
seq(n!*coeff(ser, x, n), n = 0..18); # Peter Luschny, Jan 20 2023

Mathematica
```
Table[(n+3)^n, {n, 0, 18}]
```

a(n)=(n+3)^n \\ Charles R Greathouse IV, Feb 06 2017

[(n+3)^n for n in (0..20)] # G. C. Greubel, Mar 06 2020

E.g.f. for b(n) = a(n-3): T(x) - (3/4)*T^2(x) + (1/6)*T^3(x), where T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
E.g.f.: -LambertW(-x)^3/(x^3 * (1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
With offset 1: E.g.f.: exp(T(x))^2/2 where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, May 10 2013
E.g.f.: (1/2)*d/dx (LambertW(-x)/(-x))^2. - Wolfdieter Lang, Oct 25 2022

More terms from Wesley Ivan Hurt, May 05 2014

A008785 a(n) = (n+4)^n.

Original entry on oeis.org

1, 5, 36, 343, 4096, 59049, 1000000, 19487171, 429981696, 10604499373, 289254654976, 8649755859375, 281474976710656, 9904578032905937, 374813367582081024, 15181127029874798299, 655360000000000000000, 30041942495081691894741, 1457498964228107529355264

Offset: 0

N. J. A. Sloane

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

Cf. A000169, A000272, A000312, A007778, A007830, A008786, A008787, A008788, A008789, A008790, A008791.

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

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

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

vector(20, n, (n+3)^(n-1)) \\ G. C. Greubel, Nov 09 2017

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

E.g.f.(x) for b(n) = n^(n-4) = a(n-4): T - (7/8)*T^2 + (11/36)*T^3 - (1/24)*T^4, where T = T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
E.g.f.: LambertW(-x)^4/(x^4*(1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
E.g.f.: (1/3)*d/dx(LambertW(-x)/(-x))^3. - Wolfdieter Lang, Oct 25 2022

A008788 a(n) = n^(n+2).

Original entry on oeis.org

0, 1, 16, 243, 4096, 78125, 1679616, 40353607, 1073741824, 31381059609, 1000000000000, 34522712143931, 1283918464548864, 51185893014090757, 2177953337809371136, 98526125335693359375, 4722366482869645213696

Offset: 0

N. J. A. Sloane

			G.f. = x + 16*x^2 + 243*x^3 + 4096*x^4 + 78125*x^5 + 1679616*x^6 + ...

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, A008789, A008790, A008791.

List([0..20], n-> n^(n+2)); # G. C. Greubel, Sep 11 2019

[n^(n+2): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Table[n^(n+2), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)
CoefficientList[Series[LambertW[-x] * (2*LambertW[-x]-1) / (1 + LambertW[-x])^5, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 20 2014 *)

vector(20, n, (n-1)^(n+1)) \\ G. C. Greubel, Nov 14 2017

[n^(n+2) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x): T*(1 + 2*T)*(1-T)^(-5); 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 A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^2/dx^2 {x^2/(T(x)^2*(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

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

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

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

Original entry on oeis.org

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

A090650 n^(n+6).

Original entry on oeis.org

1, 256, 19683, 1048576, 48828125, 2176782336, 96889010407, 4398046511104, 205891132094649, 10000000000000000, 505447028499293771, 26623333280885243904, 1461920290375446110677, 83668255425284801560576

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Dec 13 2003

nonn

Cf. A000312, A007778, A008788, A008789, A008790, A008791.

a:=n->mul(n,k=-5..n):seq(a(n),n=1..14); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+6),{n,0,40}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

cp's OEIS Frontend

A007778 a(n) = n^(n+1).

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A007830 a(n) = (n+3)^n.

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A008785 a(n) = (n+4)^n.

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A008788 a(n) = n^(n+2).

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

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

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