1,2

Also the number of connected transitive subtree acyclic digraphs on n vertices. - Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001

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

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

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

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

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

P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.

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.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 524.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 63.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 128.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 25, Prop. 5.3.2.

N. J. A. Sloane, Table of n, a(n) for n = 1..100

David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.

R. Castelo and A. Siebes, A characterization of moral transitive directed acyclic graph Markov models as trees, Technical Report CS-2000-44, Faculty of Computer Science, University of Utrecht.

N. Hobson, Exponential equation.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 67

F. Ruskey, Information on Rooted Trees

N. J. A. Sloane, Illustration of initial terms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

D. Zvonkine, An algebra of power series...

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Index entries for "core" sequences

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.

a(n) = A000312(n-1)*A128434(n,1)/A128433(n,1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 03 2007

A000169 := n-> n^(n-1);

spec := [A, {A=Prod(Z, Set(A))}, labeled]; [seq(combstruct[count](spec, size=n), n=1..20)];

seq(mul((n), k=2..n), n=1..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 14 2007

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

with(finance):seq(futurevalue( 1, n, n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008

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]

Table[n^(n - 1), {n, 1, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

(PARI) a(n)=if(n<1, 0, n^(n-1))

(Mupad) (1+n)^n $ n=0..21 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2007

sage: [lucas_number1(n, n, 0) for n in xrange(1, 19)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008

Cf. A000055, A000081, A000272, A000312, A007778, A007830, A008785-A008791, A055860.

See also A053506-A053509.

Cf. A002061.

Cf. A052746, A052756, A052764, A052789.

Adjacent sequences: A000166 A000167 A000168 this_sequence A000170 A000171 A000172

Sequence in context: A052514 A036776 A036777 this_sequence A055860 A152917 A112319

easy,core,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).