A008786 -id:A008786 - 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

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

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

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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