0,5

Also, number of unlabeled 2-gonal 2-trees with n 2-gons.

Equals INVERTi transform of A157904: (1, 2, 4, 8, 17, 36, 78, 170,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 08 2009]

Equals left border of triangle A157905 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 08 2009]

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.

A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.

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).

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.

D. D. Grant, The stability index of graphs, pp. 29-52 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.

F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 58 and 244.

D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.

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.

N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.

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

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

R. J. Mathar and Robert G. Wilson v, Table of n, a(n) for n = 0..1000

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

S. R. Finch, Otter's Tree Enumeration Constants

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 481

G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees

Sebastian Piec, Krzysztof Malarz and Krzysztof Kulakowski. How to count trees?, Internal. J. Modern Phys., C16 (2005) 1527-1534.

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees)., J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.

N. J. A. Sloane, Illustration of initial terms

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

Index entries for sequences related to trees

Index entries for "core" sequences

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

a(1) = 1 [o]; a(2) = 1 [o-o]; a(3) = 1 [o-o-o];

a(4) = 2 [o-o-o and o-o-o-o]

........... | ..............

........... o ..............

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

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]

# Another version: create b-file b000055.txt, from R. J. Mathar, Mar 06 2009 (Start)

A000081 := proc(n) option remember ; local d, j;

if n<=1 then n else

add ( add(d*procname(d), d=numtheory[divisors](j)) *procname(n-j), j=1..n-1)/ (n-1) ; fi ; end:

A000055 := proc(nmax) local a81, n, t, a, j ;

a81 := [seq(A000081(i), i=0..nmax)] ;

a := [] ;

for n from 0 to nmax do

if n = 0 then

t := 1+op(n+1, a81) ;

else

t := op(n+1, a81) ;

fi;

if type(n, even) then

t := t-op(1+n/2, a81)^2/2 ;

t := t+op(1+n/2, a81)/2 ;

fi;

for j from 0 to (n-1)/2 do

t := t-op(j+1, a81)*op(n-j+1, a81) ;

od:

a := [op(a), t] ;

od:

a ;

end:

# maximum b-file elements: 1000

L := A000055(1000) ;

read("transforms3") ;

LISTTOBFILE("b000055.txt", L, 0) ; # (End)

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)

(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)

Cf. A000676 (centered trees), A000677 (bicentered trees), A027416 (trees with a centroid), A102011 (trees with a bicentroid).

Cf. A000081 (rooted trees), A000272 (labeled trees), A000169 (labeled rooted trees).

Cf. A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A054581 (unlabeled 2-trees).

Main diagonal of A054924.

Cf. A157904, A157905. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 08 2009]

Sequence in context: A090344 A130131 A123465 this_sequence A006787 A000992 A036648

Adjacent sequences: A000052 A000053 A000054 this_sequence A000056 A000057 A000058

nonn,easy,nice,core

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