%I A000312 M3619 N1469
%S A000312 1,1,4,27,256,3125,46656,823543,16777216,387420489,10000000000,
%T A000312 285311670611,8916100448256,302875106592253,11112006825558016,
%U A000312 437893890380859375,18446744073709551616,827240261886336764177
%N A000312 Number of labeled mappings from n points to themselves (endofunctions): 
               n^n.
%C A000312 Also number of labeled pointed rooted trees (or vertebrates) on n nods.
%C A000312 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
%C A000312 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
%C A000312 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
%C A000312 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 A000312 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
%D A000312 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000312 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000312 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 
               Structures, Cambridge, 1998, pp. 62, 63, 87.
%D A000312 C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, 
               Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, p 146-157.
%D A000312 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.
%D A000312 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)
%H A000312 T. D. Noe, <a href="b000312.txt">Table of n, a(n) for n = 0..100</a>
%H A000312 H. Bottomley, <a href="a312.gif">Illustration of initial terms</a>
%H A000312 F. Ellermann, <a href="a001792.txt">Illustration of binomial transforms</
               a>
%H A000312 N. Hobson, <a href="http://www.qbyte.org/puzzles/p048s.html">Exponential 
               equation</a>.
%H A000312 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=36">
               Encyclopedia of Combinatorial Structures 36</a>
%H A000312 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HadamardsMaximumDeterminantProblem.html">Link to a section of The 
               World of Mathematics.</a>
%H A000312 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HankelMatrix.html">Link to a section of The World of Mathematics.</
               a>
%H A000312 D. Zvonkine, <a href="http://www.arXiv.org/abs/math.AG/0403092">An algebra 
               of power series...</a>
%H A000312 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000312 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A000312 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
%F A000312 E.g.f.: 1/(1+W(-x)), W(x) = principal branch of Lambert's function.
%F A000312 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
%F A000312 E.g.f.: 1/(1-T), where T=T(x) is Euler's tree function (see A000169).
%F A000312 a(n) = A000169(n+1)*A128433(n+1,1)/A128434(n+1,1). - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Mar 03 2007
%F A000312 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
%p A000312 A000312 := n->n^n;
%p A000312 seq(mul(n, k=1..n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 01 2007
%p A000312 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
%p A000312 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
%p A000312 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]
%p A000312 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]
%t A000312 Array[ #^# &, 16] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), May 
               01 2008)
%t A000312 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]
%t A000312 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]
%t A000312 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]
%t A000312 Table[Hyperfactorial[n]/Hyperfactorial[n - 1], {n, 0, 17}] [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
%o A000312 (PARI) a(n)=if(n<0,0,n^n)
%o A000312 (Other) sage: [log(e^(n^n))for n in xrange(0,10)]# [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 03 2009]
%Y A000312 Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791.
%Y A000312 Cf. A019538 A048993 A008279.
%Y A000312 First column of triangle A055858.
%Y A000312 A008972. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 20 2009]
%Y A000312 Sequence in context: A117280 A067040 A070271 this_sequence A050764 A052813 
               A121353
%Y A000312 Adjacent sequences: A000309 A000310 A000311 this_sequence A000313 A000314 
               A000315
%K A000312 easy,nonn,core,nice
%O A000312 0,3
%A A000312 N. J. A. Sloane (njas(AT)research.att.com).