0,3

Also number of labeled pointed rooted trees (or vertebrates) on n nods.

For n >= 1 a(n) is also the number of n X n (0,1) matrices in which each row contains exactly one entry equal to 1. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

Also the number of labeled rooted trees on (n+1) nodes such that the root is lower than its children. Also the number of alternating labeled rooted ordered trees on (n+1) nodes such that the root is lower than its children. - Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 27 2002

With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j,i) = the j-th part of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n) = sum_{i=1}^{p(n)} (n!/(prod_{j=1}^{p(i)}p(i,j)!)) * ((n!/(n-p(i)))!/(prod_{j=1}^{d(i)} m(i,j)!)) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005

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) = total number of leaves in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 leaves. - David Callan (callan(AT)stat.wisc.edu), Feb 01 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 62, 63, 87.

C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, p 146-157.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)

T. D. Noe, Table of n, a(n) for n = 0..100

H. Bottomley, Illustration of initial terms

F. Ellermann, Illustration of binomial transforms

N. Hobson, Exponential equation.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 36

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

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 "core" sequences

Index entries for sequences related to rooted trees

a(n-1) = -sum( (-1)^i * i * n^(n-1-i)*binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000

E.g.f.: 1/(1+W(-x)), W(x) = principal branch of Lambert's function.

a(n) = Sum(k>=0, C(n, k)*Stirling2(n, k)*k!) = Sum(k>=0, A008279(n, k)*A048993(n, k)) = Sum(k>=0, A019538(n, k)*A07318(n, k)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 14 2003

E.g.f.: 1/(1-T), where T=T(x) is Euler's tree function (see A000169).

a(n) = A000169(n+1)*A128433(n+1,1)/A128434(n+1,1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 03 2007

Comment on power series with denominators a(n): Let f(x) = 1 + Sum_{n = 1..oo} x^n/n^n. Then as x -> oo, f(x) ~ exp(x/e)*sqrt(2*Pi*x/e). - Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Sep 11 2008

A000312 := n->n^n;

seq(mul(n, k=1..n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2007

a:=n->mul(sum(n*(-1)^j, j=0..20), k=1..n): seq(a(n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007

a:=n->mul(denom (1/(n+1)), k=0..n): seq(a(n), n=-1..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2008

a:=n->mul(1+add(1, j=1..n), j=0..n):seq(a(n), n=-1..18); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2008]

restart:a:=n->mul(sum(1, j=0..n), k=0..n): seq(a(n), n=-1..16); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2009]

Array[ #^# &, 16] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), May 01 2008)

a[n + 1]/(a[n] E)== Limit[Product[x^(1/k), {x, n - 1 + 1/k, n, 1/k}], k -> Infinity], Replace n>=1 with a integer for solving in Mathematica 7. [From Deep Blue (dblue2001(AT)hotmail.com), Dec 28 2008]

a[n + 1]/(a[n] E)== Limit[Product[x^k, {x, n - 1 + k, n, k}], k -> 0], Replace n>=1 with a integer for solving in Mathematica 7. [From Deep Blue (dblue2001(AT)hotmail.com), Dec 28 2008]

Table[Sum[StirlingS2[n, i] i! Binomial[n, i], {i, 0, n}], {n, 0, 20}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 17 2009]

Table[Hyperfactorial[n]/Hyperfactorial[n - 1], {n, 0, 17}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

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

(Other) sage: [log(e^(n^n))for n in xrange(0, 10)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]

Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791.

Cf. A019538 A048993 A008279.

First column of triangle A055858.

A008972. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 20 2009]

Sequence in context: A117280 A067040 A070271 this_sequence A050764 A052813 A121353

Adjacent sequences: A000309 A000310 A000311 this_sequence A000313 A000314 A000315

easy,nonn,core,nice

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