%I A000169 M1946 N0771
%S A000169 1,2,9,64,625,7776,117649,2097152,43046721,1000000000,25937424601,
%T A000169 743008370688,23298085122481,793714773254144,29192926025390625,
%U A000169 1152921504606846976,48661191875666868481,2185911559738696531968
%N A000169 Number of labeled rooted trees with n nodes: n^(n-1).
%C A000169 Also the number of connected transitive subtree acyclic digraphs on n 
               vertices. - Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001
%C A000169 For any given integer k a(n) is also is the number of functions from 
               {1,2,...,n} to {1,2,...,n} such that the sum of the function values 
               is k mod n. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 16 2002
%C A000169 The n-th term of a geometric progression with first term 1 and common 
               ratio n: a(1) = 1 -> 1,1,1,1,... a(2) = 2 -> 1,2,... a(3) = 9 -> 
               1,3,9,... a(4) = 64 -> 1,4,16,64,... - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Mar 25 2004
%C A000169 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, 2, 3, ... . - Nick Hobson Nov 30 2006
%C A000169 a(n+1) is also the number of partial functions on n labeled objects. 
               - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 25 2006
%C A000169 More generally, consider the class of sequences of the form a(n)=[n*c(1)*...*c(i)]^(n-1). 
               This sequence has c(1)=1. A052746 has a(n) = [2*n]^(n-1), A052756 
               has a(n)=[3*n]^(n-1),A052764 has a(n)=[4*n]^(n-1), A052789 has a(n)=[5*n]^(n-1). 
               These sequences have a combinatorial structure like simple grammars. 
               - Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), Feb 23 2008
%D A000169 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000169 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000169 P. J. Cameron and P. Cara, Independent generating sets and geometries 
               for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
%D A000169 R. Castelo and A. Siebes, A characterization of moral transitive acyclic 
               directed graph Markov models as labeled trees, J. Statist. Planning 
               Inference, 115(1):235-259, 2003.
%D A000169 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 524.
%D A000169 Hannes Heikinheimo, Heikki Mannila and Jouni K. Seppnen, Finding Trees 
               from Unordered 01 Data, in Knowledge Discovery in Databases: PKDD 
               2006, Lecture Notes in Computer Science, Volume 4213/2006, Springer-Verlag. 
               [From N. J. A. Sloane, Jul 09 2009]
%D A000169 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 
               63.
%D A000169 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               128.
%D A000169 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               page 25, Prop. 5.3.2.
%H A000169 N. J. A. Sloane, <a href="b000169.txt">Table of n, a(n) for n = 1..100</
               a>
%H A000169 David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               A Combinatorial Derivation of the Number of Labeled Forests</a>, 
               J. Integer Seqs., Vol. 6, 2003.
%H A000169 R. Castelo and A. Siebes, <a href="http://ftp.cs.uu.nl:/pub/RUU/CS/techreps/
               CS-2000/2000-44.ps.gz">A characterization of moral transitive directed 
               acyclic graph Markov models as trees</a>, Technical Report CS-2000-44, 
               Faculty of Computer Science, University of Utrecht.
%H A000169 N. Hobson, <a href="http://www.qbyte.org/puzzles/p048s.html">Exponential 
               equation</a>.
%H A000169 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=67">
               Encyclopedia of Combinatorial Structures 67</a>
%H A000169 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/tree/RootedTree.html">
               Information on Rooted Trees</a>
%H A000169 N. J. A. Sloane, <a href="a81.html">Illustration of initial terms</a>
%H A000169 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GraphVertex.html">Link to a section of The World of Mathematics.</
               a>
%H A000169 D. Zvonkine, <a href="http://www.arXiv.org/abs/math.AG/0403092">An algebra 
               of power series...</a>
%H A000169 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A000169 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A000169 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000169 The e.g.f. T(x) = Sum_{n=1..infinity} n^(n-1)*x^n/n! satisfies T(x) = 
               x*e^T(x), so T(x) is the functional inverse of x*e^(-x). Also T(x) 
               = -LambertW(-x) where W(x) is the principal branch of Lambert's function. 
               T(x) is sometimes called Euler's tree function.
%F A000169 a(n) = A000312(n-1)*A128434(n,1)/A128433(n,1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 03 2007
%p A000169 A000169 := n-> n^(n-1);
%p A000169 spec := [A, {A=Prod(Z,Set(A))}, labeled]; [seq(combstruct[count](spec,
               size=n), n=1..20)];
%p A000169 seq(mul((n), k=2..n), n=1..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 14 2007
%p A000169 a:=n->mul(denom (1/(n+2)), k=0..n): seq(a(n), n=-1..16); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2008
%p A000169 with(finance):seq(futurevalue( 1, n, n),n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 22 2008
%p A000169 a:=n->mul(1+add(1, j=0..n),j=0..n):seq(a(n),n=-1..18);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2008]
%t A000169 Table[n^(n - 1), {n, 1, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%o A000169 (PARI) a(n)=if(n<1,0,n^(n-1))
%o A000169 (Mupad) (1+n)^n $ n=0..21 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 01 2007
%o A000169 sage: [lucas_number1(n,n,0) for n in xrange(1,19)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 16 2008
%Y A000169 Cf. A000055, A000081, A000272, A000312, A007778, A007830, A008785-A008791, 
               A055860.
%Y A000169 See also A053506-A053509.
%Y A000169 Cf. A002061.
%Y A000169 Cf. A052746, A052756, A052764, A052789.
%Y A000169 Sequence in context: A052514 A036776 A036777 this_sequence A055860 A152917 
               A112319
%Y A000169 Adjacent sequences: A000166 A000167 A000168 this_sequence A000170 A000171 
               A000172
%K A000169 easy,core,nonn,nice
%O A000169 1,2
%A A000169 N. J. A. Sloane (njas(AT)research.att.com).