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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

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

Links

• 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

Crossrefs

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]

Programs

Magma

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

Maple

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

Mathematica

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

Maxima

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

PARI

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

Sage

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

Formula

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