Search: id:A001858
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%I A001858 M1804 N0714
%S A001858 1,1,2,7,38,291,2932,36961,561948,10026505,205608536,4767440679,
%T A001858 123373203208,3525630110107,110284283006640,3748357699560961,
%U A001858 137557910094840848,5421179050350334929,228359487335194570528
%N A001858 Number of forests of trees on n labeled nodes.
%D A001858 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001858 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001858 B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
%D A001858 J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
%D A001858 L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 
               3 (1990), 574-581.
%D A001858 Wright, E. M., A relationship between two sequences, Proc. London Math. 
               Soc. (3) 17 (1967) 296-304.
%H A001858 T. D. Noe, <a href="b001858.txt">Table of n, a(n) for n=0..100</a>
%H A001858 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 A001858 Huantian Cao, <a href="http://www.cs.uga.edu/~rwr/STUDENTS/hcao.html">
               AutoGF: An Automated System to Calculate Coefficients of Generating 
               Functions</a>.
%H A001858 J. Pitman, <a href="http://www.stat.berkeley.edu/users/pitman/457.pdf">
               Coalescent Random Forests</a>, J. Combin. Theory, A85 (1999), 165-193.
%H A001858 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               132
%F A001858 E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem 
               of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane (njas(AT)research.att.com), 
               May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
%F A001858 E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{ n>=1} n^(n-1)*x^n/n! is 
               Euler's tree function (see A000169). - Len Smiley (smiley(AT)math.uaa.alaska.edu), 
               Dec 12 2001
%F A001858 Shifts 1 place left under the hyperbinomial transform (cf. A088956). 
               - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 03 2003
%e A001858 a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>1. 
               [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
%p A001858 exp(x+x^2+add(n^(n-2)*x^n/n!,n=3..50));
%p A001858 a:= proc(n) option remember; if n=0 then 1 else add (binomial (n-1, j) 
               *(j+1)^(j-1) *a(n-1-j), j=0..n-1) fi end: seq (a(n), n=0..20); [From 
               Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
%o A001858 (PARI) a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n-1,n-m-j)*n^(n-m-j)*(m+j)!/
               (-2)^j)/m!))
%Y A001858 Cf. A088956. Row sums of A138464.
%Y A001858 Sequence in context: A145159 A084552 A094664 this_sequence A000366 A106211 
               A014058
%Y A001858 Adjacent sequences: A001855 A001856 A001857 this_sequence A001859 A001860 
               A001861
%K A001858 nonn,easy,eigen
%O A001858 0,3
%A A001858 N. J. A. Sloane (njas(AT)research.att.com) and Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A001858 PARI code and more terms from Michael Somos, Aug 22, 2002.

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