Search: id:A000055
Results 1-1 of 1 results found.

%I A000055 M0791 N0299
%S A000055 1,1,1,1,2,3,6,11,23,47,106,235,551,1301,3159,7741,19320,48629,123867,
%T A000055 317955,823065,2144505,5623756,14828074,39299897,104636890,279793450,
%U A000055 751065460,2023443032,5469566585,14830871802,40330829030,109972410221
%N A000055 Number of trees with n unlabeled nodes.
%C A000055 Also, number of unlabeled 2-gonal 2-trees with n 2-gons.
%C A000055 Equals INVERTi transform of A157904: (1, 2, 4, 8, 17, 36, 78, 170,...). 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 08 2009]
%C A000055 Equals left border of triangle A157905 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 08 2009]
%D A000055 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 
               Structures, Camb. 1998, p. 279.
%D A000055 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
%D A000055 A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 
               266-268.
%D A000055 A. Cayley, On the analytical forms called trees, with application to 
               the theory of chemical combinations, Reports British Assoc. Advance. 
               Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 459).
%D A000055 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
%D A000055 D. D. Grant, The stability index of graphs, pp. 29-52 of Combinatorial 
               Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 
               403, 1974.
%D A000055 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 526.
%D A000055 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
%D A000055 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 58 and 244.
%D A000055 D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.
%D A000055 Elena V. Konstantinova and Maxim V. Vidyuk, "Discriminating tests of 
               information and topological indices. Animals and trees", J. Chem. 
               Inf. Comput. Sci., (2003), vol. 43, 1860-1871. See Table 15, column 
               1 on page 1868.
%D A000055 N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 
               14 (2001), 93-115.
%D A000055 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A000055 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               138.
%D A000055 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000055 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000055 R. J. Mathar and Robert G. Wilson v, <a href="b000055.txt">Table of n, 
               a(n) for n = 0..1000</a>
%H A000055 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000055 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/otter/otter.html">
               Otter's Tree Enumeration Constants</a>
%H A000055 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               481
%H A000055 G. Labelle, C. Lamathe and P. Leroux, <a href="http://arXiv.org/abs/math.CO/
               0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>
%H A000055 Sebastian Piec, Krzysztof Malarz and Krzysztof Kulakowski. <a href="http:/
               /arXiv.org/abs/cond-mat/0501594">How to count trees?</a>, Internal. 
               J. Modern Phys., C16 (2005) 1527-1534.
%H A000055 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent 
               Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A000055 N. J. A. Sloane, <a href="a55.gif">Illustration of initial terms</a>
%H A000055 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Tree.html">Link to a section of The World of Mathematics.</a>
%H A000055 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A000055 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000055 G.f.: A(x) = 1 + T(x)-T^2(x)/2+T(x^2)/2, where T(x) = x + x^2 + 2*x^3 
               + ... is g.f. for A000081
%e A000055 a(1) = 1 [o]; a(2) = 1 [o-o]; a(3) = 1 [o-o-o];
%e A000055 a(4) = 2 [o-o-o and o-o-o-o]
%e A000055 ........... | ..............
%e A000055 ........... o ..............
%p A000055 G000055 := series(1+G000081-G000081^2/2+subs(x=x^2,G000081)/2,x,31); 
               A000055 := n->coeff(G000055,x,n); # where G000081 is g.f. for A000081 
               starting with n=1 term
%p A000055 b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), 
               k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), 
               j=1..iquo(n,k)) end: B:= proc(n) option remember; unapply (add (b(k)*x^k, 
               k=1..n),x) end: a:= n-> coeff (series (1+ B(n)(x)- (B(n)(x))^2/2+ 
               B(n)(x^2)/2, x=0, n+1),x,n): seq (a(n), n=0..32); [From Alois P. 
               Heinz (heinz(AT)hs-heilbronn.de), Aug 21 2008]
%p A000055 # Another version: create b-file b000055.txt, from R. J. Mathar, Mar 
               06 2009 (Start)
%p A000055 A000081 := proc(n) option remember ; local d, j;
%p A000055 if n<=1 then n else
%p A000055 add ( add(d*procname(d),d=numtheory[divisors](j)) *procname(n-j), j=1..n-1)/ 
               (n-1) ; fi ; end:
%p A000055 A000055 := proc(nmax) local a81,n,t,a,j ;
%p A000055 a81 := [seq(A000081(i),i=0..nmax)] ;
%p A000055 a := [] ;
%p A000055 for n from 0 to nmax do
%p A000055 if n = 0 then
%p A000055 t := 1+op(n+1,a81) ;
%p A000055 else
%p A000055 t := op(n+1,a81) ;
%p A000055 fi;
%p A000055 if type(n,even) then
%p A000055 t := t-op(1+n/2,a81)^2/2 ;
%p A000055 t := t+op(1+n/2,a81)/2 ;
%p A000055 fi;
%p A000055 for j from 0 to (n-1)/2 do
%p A000055 t := t-op(j+1,a81)*op(n-j+1,a81) ;
%p A000055 od:
%p A000055 a := [op(a),t] ;
%p A000055 od:
%p A000055 a ;
%p A000055 end:
%p A000055 # maximum b-file elements: 1000
%p A000055 L := A000055(1000) ;
%p A000055 read("transforms3") ;
%p A000055 LISTTOBFILE("b000055.txt",L,0) ; # (End)
%t A000055 s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ] ]; a[ 1 
               ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); 
               Table[ a[ i ]-Sum[ a[ j ]a[ i-j ], {j, 1, i/2} ]+If[ OddQ[ i ], 0, 
               a[ i/2 ](a[ i/2 ]+1)/2 ], {i, 1, 50} ] (from Robert A. Russell)
%o A000055 (PARI) a(n)=local(A, A1,an,i,t); if(n<2,n>=0,an=Vec(A=A1=1+O('x^n)); 
               for(m=2,n,i=m\2; an[m]=sum(k=1,i,an[k]*an[m-k])+(t=polcoeff(if(m%2,
               A*=(A1-'x^i)^-an[i],A),m-1))); t+if(n%2==0,binomial(-polcoeff(A,i-1),
               2))) (from Michael Somos)
%Y A000055 Cf. A000676 (centered trees), A000677 (bicentered trees), A027416 (trees 
               with a centroid), A102011 (trees with a bicentroid).
%Y A000055 Cf. A000081 (rooted trees), A000272 (labeled trees), A000169 (labeled 
               rooted trees).
%Y A000055 Cf. A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 
               4-trees), A054581 (unlabeled 2-trees).
%Y A000055 Main diagonal of A054924.
%Y A000055 Cf. A157904, A157905. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 
               08 2009]
%Y A000055 Sequence in context: A090344 A130131 A123465 this_sequence A006787 A000992 
               A036648
%Y A000055 Adjacent sequences: A000052 A000053 A000054 this_sequence A000056 A000057 
               A000058
%K A000055 nonn,easy,nice,core
%O A000055 0,5
%A A000055 N. J. A. Sloane (njas(AT)research.att.com).

Search completed in 0.002 seconds