• If the Internet is slow, or if you are using a device that can send electronic mail but cannot access the Internet, it is possible to consult the On-Line Encyclopedia of Integer Sequences via email.
• Simply send an email message to
  sequences@research.att.com

  saying (for example):

  lookup 1 3 16 125 1296 16807 262144
  lookup 1 2 5 14 42 132 429 1430 4862
  

• The message may include up to 10 such requests.
  The entries should be separated by spaces rather than commas.
  The "Subject" line should say "None" or be left blank.
  Sending an empty message will produce the "Help" file.
• The reply normally comes back almost at once.
• Here for example is the response to the first of the two requests above. (Of course exactly how this appears will depend on the mail program that you are using.)

From sequences-reply@research.att.com  Mon Jan 23 22:59:19 2006
Date: Mon, 23 Jan 2006 22:59:19 -0500 (EST)
From: sequences-reply@research.att.com
Subject: Reply from On-Line Encyclopedia of Integer Sequences
 
Matches (up to a limit of 50) found for  1 3 16 125 1296 16807 262144  :

%I A000272 M3027 N1227
%S A000272 1,1,3,16,125,1296,16807,262144,4782969,100000000,2357947691,61917364224,
%T A000272 1792160394037,56693912375296,1946195068359375,72057594037927936,
%U A000272 2862423051509815793,121439531096594251776,5480386857784802185939
%N A000272 Number of labeled trees on n nodes: n^(n-2).
%C A000272 Number of spanning trees in complete graph K_n on n labeled nodes.
%C A000272 Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001, observes that
               n^(n-2) is also the number of transitive subtree acyclic digraphs
               on n-1 vertices.
%C A000272 a(n) is also the number of ways of expressing an n-cycle in the
               symmetric group S_n as a product of n-1 transpositions. - Dan
               Fux (danfux(AT)my-deja.com), Apr 12 2001
%C A000272 Also counts parking functions, noncrossing partitions, critical
               configurations of the chip firing game, allowable pairs sorted
               by a priority queue [Hamel].
%C A000272 a(n+1) = sum( i * n^(n-1-i) * binomial(n, i), i=1..n) - Yong Kong
               (ykong(AT)curagen.com), Dec 28 2000
%D A000272 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag,
               Berlin, 1999; see p. 142.
%D A000272 M. D. Atkinson and R. Beals, Priority queues and permutations,
               SIAM J. Comput. 23 (1994), 1225-1230.
%D A000272 N. L. Biggs, Chip-firing and the critical group of a graph,
               J. Algeb. Combin., 9 (1999), 25-45.
%D A000272 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 51.
%D A000272 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 A000272 J. Denes, The representation of a permutation as the product
               of a minimal number of transpositions ..., Pub. Math. Inst.
               Hung. Acad. Sci., 4 (1959), 63-70.
%D A000272 J. Gilbey and L. Kalikow, Parking functions, valet functions
               and priority queues, Discrete Math., 197 (1999), 351-375.
%D A000272 M. Golin and S. Zaks, Labeled trees and pairs of input-output
               permutations in priority queues, Theoret. Comput. Sci.,
               205 (1998), 99-114.
%D A000272 I. P. Goulden and S. Pepper, Labeled trees and factorizations
               of a cycle into transpositions, Discrete Math., 113 (1993), 263-268.
%D A000272 I. P. Goulden and A. Yong, Tree-like properties of cycle
               factorizations, J. Combin. Theory, A 98 (2002), 106-117.
%D A000272 A. M. Hamel, Priority queue sorting and labeled trees,
               Annals Combin., 7 (2003), 49-54.
%D A000272 D. M. Jackson - Some Combinatorial Problems Associated with
               Products of Conjugacy Classes of the Symmetric Group,
               Journal of Combinatorial Theory, Seies A, 49 363-369(1988).
%D A000272 S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of
               the Giant Component, Random Structures and Algorithms
               Vol. 4 (1993), 233-358.
%D A000272 L. Kalikow, Symmetries in trees and parking functions,
               Discrete Math., 256 (2002), 719-741.
%D A000272 J. H. van Lint and R. M. Wilson, A Course in Combinatorics,
               Cambridge Univ. Press, 1992.
%D A000272 F. McMorris and F. Harary (1992), Subtree acyclic digraphs,
               Ars Comb., vol. 34.
%D A000272 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)
%D A000272 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 128.
%D A000272 M. P. Schutenberger, On an Enumeration Problem, Journal of
               Combinatorial Theory 4, 219-221 (1968). [A 1-1 correspondence
               between maps under permutations and acyclics maps.]
%D A000272 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999;
               see page 25, Prop. 5.3.2.
%D A000272 R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull.
               Amer. Math. Soc., 40 (2003), 55-68.
%H A000272 Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions.
%H A000272 R. Castelo and A. Siebes, A characterization of moral transitive directed acyclic graph  ..., 
               Report CS-2000-44, Department of Computer Science, Univ. Utrecht.
%H A000272 S. Coulomb and M. Bauer, On vertex covers, matchings, and random trees
%H A000272 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 78
%H A000272 C. Lamathe, The Number of Labeled k-Arch Graphs,
               Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.1.
%H A000272 S. Ramanujan, Question 738, J. Ind. Math. Soc.
%H A000272 E. W. Weisstein, Link to a section of The World of Mathematics.
%H A000272 D. Zeilberger, The n^(n-2)-th Proof Of The Formula For The Number Of Labeled Trees
%H A000272 D. Zeilberger, Yet Another Proof For The Enumeration Of Labeled Trees
%H A000272 D. Zvonkine, An algebra of power series...
%H A000272 Index entries for sequences related to trees
%H A000272 Index entries for "core" sequences
%F A000272 E.g.f.: ((W(-x)/x)^2)/(1+W(-x)), W(x): Lambert's function (principal branch).
%F A000272 E.g.f.: T - (1/2)T^2; where T=T(x) is Euler's tree function (see A000169).
                - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 19 2001
%F A000272 Number of labeled k-trees on n nodes is binomial(n,k) * (k(n-k)+1)^(n-k-2).
%p A000272 A000272 := n->n^(n-2); [ seq(n^(n-2), n=1..20) ];
%o A000272 (PARI) a(n)=if(n<1,0,n^(n-2))
%Y A000272 Cf. A000055, A000169, A000312, A007778, A007830, A008785-A008791.
                a(n)= A033842(n-1, 0) (first column of triangle).
%Y A000272 Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362
                (labeled 3-trees), A036506 (labeled 4-trees), A000055
                (unlabeled trees), A054581 (unlabeled 2-trees).
%Y A000272 Cf. A097170.
%Y A000272 Sequence in context: A090135 A000950 A000951 this_sequence A088358 A082161 A051921
%Y A000272 Adjacent sequences: A000269 A000270 A000271 this_sequence A000273 A000274 A000275
%K A000272 easy,nonn,core,nice
%O A000272 1,3
%A A000272 N. J. A. Sloane (njas(AT)research.att.com).

Search was carried out on Mon Jan 23 22:59:16 EST 2006

o  Take a look at my web page which does lookups "online"!  Go to:
     http://www.research.att.com/~njas/sequences/
o  The whole sequence table is also visible there, as well as
     an explanation of the symbols used in the table.
o  If your sequence was not in the table,
     please send it to me using the submission form on the web page!
o  There is a second sequence server (superseeker@research.att.com)
   that tries hard to find an explanation.  Only 1 request per person
   per hour please.
o  If the word "lookup" does not appear you will be sent the help file.

Sequentially yours, The On-Line Encyclopedia of Integer Sequences,
N. J. A. Sloane, AT&T Research, Florham Park NJ 07932-0971 USA njas@research.att.com

• Note that the reply shows the sequence represented in the "internal" format used in the database, rather than in the "beautified" format provided by the web pages. (Regular users prefer this format.)
• The internal format is described in the file eishelp1.html.
• Superseeker, a much more sophisticated program for analyzing sequences, and also accessed via email, is described in the next demonstration.

Click the single right arrow to go to the next demonstration page, or the single left arrow to return to the previous page.