A007778 - OEIS

%I #79 Feb 18 2025 14:26:45

%S 0,1,8,81,1024,15625,279936,5764801,134217728,3486784401,100000000000,

%T 3138428376721,106993205379072,3937376385699289,155568095557812224,

%U 6568408355712890625,295147905179352825856,14063084452067724991009,708235345355337676357632

%N a(n) = n^(n+1).

%C Number of edges of the complete bipartite graph of order n+n^n, K_n,n^n. - _Roberto E. Martinez II_, Jan 07 2002

%C 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

%C 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

%C 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

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

%H Vincenzo Librandi, <a href="/A007778/b007778.txt">Table of n, a(n) for n = 0..200</a>

%H Nick Hobson, <a href="https://web.archive.org/web/20160413232742/http://www.qbyte.org/puzzles/p048s.html">Exponential equation</a>.

%H Yidong Sun and Jujuan Zhuang, <a href="http://arxiv.org/abs/1007.1339">lambda-factorials of n</a>, arXiv:1007.1339 [math.CO], 2010. - _Peter Luschny_, Jul 09 2010

%F E.g.f.: -W(-x)/(1 + W(-x))^3, W(x) Lambert's function (principal branch).

%F a(n) = Sum_{k=0..n} binomial(n,k)*A000166(k+1)*(n+1)^(n-k). - _Peter Luschny_, Jul 09 2010

%F See A008517 and A134991 for similar e.g.f.s. and A048993. - _Tom Copeland_, Oct 03 2011

%F 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

%F a(n) = n*A000312(n). - _R. J. Mathar_, Jan 12 2017

%F Sum_{n>=2} 1/a(n) = A135608. - _Amiram Eldar_, Nov 17 2020

%p seq( n^(n+1), n=0..20); # _G. C. Greubel_, Mar 05 2020

%t Table[n^(n+1), {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Oct 01 2008 *)

%o (Magma) [n^(n+1):n in [0..20]]; // _Vincenzo Librandi_, Jan 03 2012

%o (Maxima) A007778[n]:=n^(n+1)$

%o makelist(A007778[n],n,0,30); /* _Martin Ettl_, Oct 29 2012 */

%o (PARI) vector(21, n, my(m=n-1); m^(m+1)) \\ _G. C. Greubel_, Mar 05 2020

%o (Sage) [n^(n+1) for n in (0..20)] # _G. C. Greubel_, Mar 05 2020

%Y Cf. A000169, A000272, A000312, A007830, A008785, A008786, A008787, A008788, A008789, A008790, A008791, A135608.

%Y Essentially the same as A065440.

%Y Cf. A061250, A143857. [From _Reinhard Zumkeller_, Jul 23 2010]

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_