A005195 -id:A005195 - cp's OEIS frontend

A123423 Duplicate of A005195.

1, 2, 3, 6, 10, 20, 37, 76, 153, 329

Offset: 1

dead

A144978 A005195(n) - A005195(n-2).

1, 1, 2, 4, 7, 14, 27, 56, 116, 253, 557, 1272, 2948, 6998, 16856, 41306, 102449, 257294, 652566, 1670679, 4311109, 11206278, 29316294, 77144518, 204072054, 542446974, 1448230644, 3882179984, 10445521740, 28202158173, 76387639678

Offset: 1

Washington Bomfim, Sep 28 2008

nonn

a(n) is the number of forests with n unlabeled nodes without trees of order 2. This follows from the fact that for n>=2 A005195(n-2) counts the forests of order n with one or more trees of order 2.
The unique forest of order 1 is an isolated vertex, so a(1)=1. For n>=2, a(n) - a(n-1) counts forests of order n with trees of order >=3.
A005195(n) - A005195(n-3) counts forests of order n without trees of order 3.

			a(5) = 7 since the only options are: 3 trees of order 5; 2 forests composed by trees of orders 4 and l; one forest with trees of orders [3 1 1]; and one forest with five isolated nodes.

Cf. A005195, A000055(trees).

A000055 Number of trees with n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 48629, 123867, 317955, 823065, 2144505, 5623756, 14828074, 39299897, 104636890, 279793450, 751065460, 2023443032, 5469566585, 14830871802, 40330829030, 109972410221, 300628862480, 823779631721, 2262366343746, 6226306037178

Offset: 0

N. J. A. Sloane

Also, number of unlabeled 2-gonal 2-trees with n-1 2-gons, for n>0. [Corrected by Andrei Zabolotskii, Jul 29 2025]
Main diagonal of A054924.
Left border of A157905. - Gary W. Adamson, Mar 08 2009
From Robert Munafo, Jan 24 2010: (Start)
Also counts classifications of n items that require exactly n-1 binary partitions; see Munafo link at A005646, also A171871 and A171872.
The 11 trees for n = 7 are illustrated at the Munafo web link.
Link to A171871/A171872 conjectured by Robert Munafo, then proved by Andrew Weimholt and Franklin T. Adams-Watters on Dec 29 2009. (End)
This is also "Number of tree perfect graphs on n nodes" [see Hougardy]. - N. J. A. Sloane, Dec 04 2015
For n > 0, a(n) is the number of ways to arrange n-1 unlabeled non-intersecting circles on a sphere. - Vladimir Reshetnikov, Aug 25 2016
All trees for n=1 through n=12 are depicted in Chapter 1 of the Steinbach reference. On p. 6 appear encircled two trees (with n=10) which seem inequivalent only when considered as ordered (planar) trees. Earlier instances of such possibly (in)equivalent trees could appear from n=6 on (and from n=9 on without equivalence modulo plane symmetry) but are not drawn separately there. - M. F. Hasler, Aug 29 2017

			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
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 55.
N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
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.
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.
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
Jeremy C. Adcock, Sam Morley-Short, Joshua W. Silverstone, and Mark G. Thompson, Hard limits on the postselectability of optical graph states, arXiv:1806.03263 [quant-ph], 2018.
H. R. Afshar, E. A. Bergshoeff, and W. Merbis, Interacting spin-2 fields in three dimensions, arXiv preprint arXiv:1410.6164 [hep-th], 2014-2015.
Ruggero Bandiera and Florian Schaetz, Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907 [math.CO], 2017. Mentions this sequence.
Madeleine Burkhart and Joel Foisy, Enumerating spherical n-links, Involve, Vol. 11 (2018), No. 2, 195-206.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
CombOS - Combinatorial Object Server, Generate graphs
Benjamin Côté, Hélène Cossette, and Etienne Marceau, Centrality and topology properties in a tree-structured Markov random field, arXiv:2410.20240 [math.ST], 2024. See p. 18.
Antoine Dailly and Tuomo Lehtilä, Reconstructing graphs with subgraph compositions, arXiv:2504.00169 [math.CO], 2025. See p. 28.
R. Ferrer-i-Cancho, Non-crossing dependencies: least effort, not grammar, arXiv preprint arXiv:1411.2645 [cs.CL], 2014.
S. R. Finch, Otter's Tree Enumeration Constants
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 481.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45 and arXiv:1208.5993 [math.CO], 2012.
Xiangrui Gao, Song He, and Yong Zhang, Labelled tree graphs, Feynman diagrams and disk integrals arxiv:1708.08701 [hep-th], 2017, see p. 4.
Ira M. Gessel, Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees, arXiv:2305.03157 [math.CO], 2023.
D. D. Grant, The stability index of graphs, pp. 29-52 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974. Gives first 45 terms.
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021. Mentions this sequence p. 3.
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
House of Graphs, Trees
Andrew Jobbings, Enumerating nets, Preprint 2015.
Mithra Karamchedu and Lucas Bang, Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism, arXiv:2508.13480 [cs.DS], 2025. See p. 4.
Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871. See Table 15, column 1 on page 1868.
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.
Steve Lawford and Yll Mehmeti, Cliques and a new measure of clustering: with application to U.S. domestic airlines, arXiv:1806.05866 [cs.SI], 2018.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Gang Li, Generation of Rooted Trees and Free Trees, Comp. Sci. MSc thesis paper, 1996.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Robert Munafo, Relation to Tree Graphs
R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.
Sebastian Piec, Krzysztof Malarz and Krzysztof Kulakowski. How to count trees?, arXiv:cond-mat/0501594 [cond-mat.stat-mech], 2005; Int. J. Modern Phys., C16 (2005) 1527-1534.
N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
J. M. Plotkin and J. W. Rosenthal, How to obtain an asymptotic expansion of a sequence from an analytic identity satisfied by its generating function, J. Austral. Math. Soc. (Series A), Vol. 56 (1994), 131-143.
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.
R. W. Robinson, Letter to N. J. A. Sloane, Jul 29 1980
Yaniv Sadeh, Haim Kaplan, and Uri Zwick, Search Trees on Trees via LP, arXiv:2501.17563 [cs.DS], 2025. See p. 12.
Sage, Common Graphs
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
N. J. A. Sloane, Illustration of initial terms
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Overview of the following parts: Cover, Front matter, Chapter 1: Trees, Trees (cont'd: pt.2), Trees (cont'd: pt.3), Chapter 2: Centers and Centroids, Chap.2 (cont'd), Chapter 3: Random Trees, Chapter 4: Rooted Trees, Chapter 5: Homeomorphically Irreducible Trees, Chapter 6: Tables (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Ivan Stošić and Ivan Damnjanović, An efficient algorithm for generating transmission irregular trees, arXiv:2502.15453 [cs.DM], 2025. See p. 11.
Tanay Wakhare, Eric Wityk, and Charles R. Johnson. The proportion of trees that are linear, Discrete Mathematics 343.10 (2020): 112008. Also arXiv:1901.08502 [math.CO], 2019-2020. See Tables 1 and 2 (but beware errors).
Eric Weisstein's World of Mathematics, Tree
Pascal Welke, Tamás Horváth, and Stefan Wrobel, Probabilistic and exact frequent subtree mining in graphs beyond forests, Machine Learning (2019), 1-28.
Robert Alan Wright, Bruce Richmond, Andrew Odlyzko, and Brendan D. McKay, Constant Time Generation of Free Trees, SIAM Journal of Computing, vol. 15, no. 2, pp. 540-548, (1986) [preprint].
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000676 (centered trees), A000677 (bicentered trees), A027416 (trees with a centroid), A102011 (trees with a bicentroid), A034853 (refined by diameter), A238414 (refined by maximum vertex degree).
Cf. A000081 (rooted trees), A000272 (labeled trees), A000169 (labeled rooted trees), A212809 (radius of convergence).
Cf. A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A054581 (unlabeled 2-trees).
Cf. A157904, A157905, A005195 (Euler transform = forests), A095133 (multisets).
Column 0 of A335362 and A034799.
Related to A005646; see A171871 and A171872.
Cf. also A000088, A008406, A051491, A086308.

import Data.List (generic_index)
import Math.OEIS (getSequenceByID)
triangle x = (x * x + x) `div` 2
a000055 n = let {r = genericIndex (fromJust (getSequenceByID "A000081")); (m, nEO) = divMod n 2}
            in  r n - sum (zipWith (*) (map r [0..m]) (map r [n, n-1..]))
                + (1-nEO) * (triangle (r m + 1))
-- Walt Rorie-Baety, Jun 12 2021

N := 30; P := PowerSeriesRing(Rationals(),N+1); f := func< A | x*&*[Exp(Evaluate(A,x^k)/k) : k in [1..N]]>; G := x; for i in [1..N] do G := f(G); end for; G000081 := G; G000055 := 1 + G - G^2/2 + Evaluate(G,x^2)/2; A000055 := Eltseq(G000055); // Geoff Baileu (geoff(AT)maths.usyd.edu.au), Nov 30 2009

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
with(numtheory): b:= proc(n) option remember; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> `if`(n=0, 1, b(n) -(add(b(k) *b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2):
seq(a(n), n=0..50);
# Alois P. Heinz, Aug 21 2008
# Program to create b-file b000055.txt:
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, i ;
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:
L := A000055(1000) ;
#  R. J. Mathar, Mar 06 2009

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}] (* Robert A. Russell *)
b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[d*b[d]*b[n-j], {j, 1, n-1}, {d, Divisors[j]}]/(n-1); a[0] = 1; a[n_] := b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

{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)))}; /* Michael Somos */

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d * A[d]) * A[n-k+1] ) );
A000081=concat([0], A);
H(t)=subst(Ser(A000081, 't), 't, t);
x='x+O('x^N);
Vec( 1 + H(x) - 1/2*( H(x)^2 - H(x^2) ) )
\\ Joerg Arndt, Jul 10 2014

# uses function from A000081
def A000055(n): return 1 if n == 0 else A000081(n)-sum(A000081(i)*A000081(n-i) for i in range(1,n//2+1)) + (0 if n % 2 else (A000081(n//2)+1)*A000081(n//2)//2) # Chai Wah Wu, Feb 03 2022

[len(list(graphs.trees(n))) for n in range(16)] # Peter Luschny, Mar 01 2020

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 the g.f. for A000081.
a(n) ~ A086308 * A051491^n * n^(-5/2). - Vaclav Kotesovec, Jan 04 2013
a(n) = A000081(n) - A217420(n+1), n > 0. - R. J. Mathar, Sep 19 2016
a(n) = A000676(n) + A000677(n). - R. J. Mathar, Aug 13 2018
a(n) = A000081(n) - (Sum_{1<=i<=j, i+j=n} A000081(i)*A000081(j)) + (1-(-1)^(n-1)) * binomial(A000081(n/2)+1,2) / 2 [Li, equation 4.2]. - Walt Rorie-Baety, Jul 05 2021

A001858 Number of forests of trees on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616

Offset: 0

N. J. A. Sloane and Simon Plouffe

The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
Number of labeled acyclic graphs on n vertices. The unlabeled version is A005195. The covering case is A105784, connected A000272. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of the a(4) = 38 forests:
  {}  {12}  {12,13}  {12,13,14}
      {13}  {12,14}  {12,13,24}
      {14}  {12,23}  {12,13,34}
      {23}  {12,24}  {12,14,23}
      {24}  {12,34}  {12,14,34}
      {34}  {13,14}  {12,23,24}
            {13,23}  {12,23,34}
            {13,24}  {12,24,34}
            {13,34}  {13,14,23}
            {14,23}  {13,14,24}
            {14,24}  {13,23,24}
            {14,34}  {13,23,34}
            {23,24}  {13,24,34}
            {23,34}  {14,23,24}
            {24,34}  {14,23,34}
                     {14,24,34}
(End)

B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.o
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See pp. 32-33.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct 19 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1975
John Riordan and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
Index entries for sequences related to forests

The connected case is A000272, rooted A000169.
The unlabeled version is A005195, connected A000055.
The covering case is A105784, unlabeled A144958.
Row sums of A138464.
For triangles instead of cycles we have A213434, covering A372168.
For a unique cycle we have A372193, covering A372195.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A006572, A006573, A088956.
Cf. A053530, A054548, A137916, A263340, A322661, A367863, A372176.

exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F,x,51):
seq(coeff(S,x,j)*j!, j=0..50); # Robert Israel, May 21 2015

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[t-t^2/2],{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)

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!)) /* Michael Somos, Aug 22 2002 */

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, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
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, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008

More terms from Michael Somos, Aug 22 2002

A140637 Number of unlabeled graphs of positive excess with n nodes.

Original entry on oeis.org

0, 0, 0, 2, 15, 110, 936, 12073, 273972, 12003332, 1018992968, 165091159269, 50502031331411, 29054155657134165, 31426485969804026075, 64001015704527557101231, 245935864153532932681481794, 1787577725145611700547871854870, 24637809253125004524383007473440146

Offset: 1

Washington Bomfim, May 21 2008

nonn

We can find in "The Birth of the Giant Component" p. 53, see the link, the following: "The excess of a graph or multigraph is the number of edges plus the number of acyclic components, minus the number of vertices."
If G has just one complex component with 4 nodes, the "non-complex part" of G can be,
a) One forest of order 4. There are 6 forests, so 2*6=12 graphs.
b) One triangle and one isolated vertex, or 2*1=2 graphs.
c) One unicyclic graph of order 4, so 2*2=4 graphs.
Also the number of unchoosable unlabeled graphs with up to n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The labeled version is A367867, covering A367868, connected A140638. - Gus Wiseman, Feb 13 2024

			Below we show that a(8) = 12073. Note that A140636(n) is the number of connected graphs of positive excess with n nodes.
Let G be a disconnected graph of positive excess with 8 nodes. In this case, G has one or two complex components. We have 3 graphs of order 8 with two complex components. One of those graphs is depicted in the figure below:
  O---O...O---O
  |\..|...|\./|
  |.\.|...|.X.|
  |..\|...|/.\|
  O---O...O---O
If G has one complex component with 5 nodes, the non-complex part of G can be,
a) One forest of order 3. There are 3 forests, so A140636(5) * 3 = 39 graphs.
b) One triangle, so A140636(5) = 13 graphs.
If G has one complex component with 6 nodes, the non-complex part of G is a forest of order 2. There are 2 forests. We have A140636(6) * 2, or 186 graphs.
If G has one complex component with 7 nodes, the non-complex part of G is one isolated vertex. We have A140636(7), or 809 graphs.
Finally if G is connected, we have A140636(8), or 11005 graphs.
The total is 3 + 12 + 2 + 4 + 39 + 13 + 186 + 809 + 11005 = 12073.

Svante Janson, Donald E. Knuth, Tomasz Łuczak and Boris Pittel, The Birth of the Giant Component.

Cf. A000088, A005195, A001429.
The labeled complement is A133686, covering A367869, connected A129271.
The complement is A134964, connected A005703.
The connected covering case is A140636.
The labeled version is A367867, covering A367868, connected A140638.
Set-systems not of this type are A367902, ranks A367906.
Set-systems of this type are A367903, ranks A367907.
For set-systems we have A368094, complement A368095.
For multiset partitions we have A368097, complement A368098.
Factorizations of this type are A368413, complement A368414.
Cf. A001349, A002494, A006125, A055621, A367769, A367770, A367901.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]],Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

a(n) = A000088(n) - A134964(n).

A005470 Number of unlabeled planar simple graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 11, 33, 142, 822, 6966, 79853, 1140916, 18681008, 333312451

Offset: 0

N. J. A. Sloane

Euler transform of A003094. - Christian G. Bower

			a(2) = 2 since o o and o-o are the two planar simple graphs on two nodes.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Trotter, ed., Planar Graphs, Vol. 9, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., 1993.
Turner, James; Kautz, William H. A survey of progress in graph theory in the Soviet Union. SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 19. - N. J. A. Sloane, Apr 08 2014
Vetukhnovskii, F. Ya. "Estimate of the Number of Planar Graphs." In Soviet Physics Doklady, vol. 7, pp. 7-9. 1962. - From N. J. A. Sloane, Apr 08 2014
R. J. Wilson, Introduction to Graph Theory. Academic Press, NY, 1972, p. 162.

G. Brinkmann, and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007) 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
E. Friedman, Illustration of small graphs
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Planar Graph
Index entries for "core" sequences

Cf. A003094 (connected planar graphs), A034889, A039735 (planar graphs by nodes and edges).

Cf. A126201.

A003094 = Cases[Import["https://oeis.org/A003094/b003094.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A003094] (* Jean-François Alcover, Apr 25 2013, updated Mar 17 2020 *)

n=8 term corrected and n=9..11 terms calculated by Brendan McKay
Terms a(0) - a(10) confirmed by David Applegate and N. J. A. Sloane, Mar 09 2007
a(12) added by Vaclav Kotesovec after A003094 (computed by Brendan McKay), Dec 06 2014

A033995 Number of bipartite graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 35, 88, 303, 1119, 5479, 32303, 251135, 2527712, 33985853, 611846940, 14864650924, 488222721992, 21712049275198, 1308300679611469, 106897965189674291, 11852113048215107822, 1784730721403509209215, 365323537513403184463273

Offset: 0

Ronald C. Read

All bipartite graphs are perfect. - Falk Hüffner, Nov 27 2015

EULER transform of A005142 [1, 1, 1, 3, 5, 17, ...] is [1, 2, 3, 7, 13, ...]. - Michael Somos, May 13 2019

			For n=1: o; n=2: o o, o-o; n=3: o o o, o o-o, o-o-o; n=4: o o o o, o o o-o, o-o o-o, o o-o-o, o-o-o-o, K_{2,2}, K_{3,1}. - _Michael Somos_, May 13 2019

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

Andrew Howroyd, Table of n, a(n) for n = 0..50
CombOS - Combinatorial Object Server, Generate graphs.
P. Erdős, D. J. Kleitman, and B. L. Rothschild, Asymptotic enumeration of k_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie, (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 19-27. Accad. Naz. Lincei, Rome.
P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49-57.
S. Hougardy, Home Page.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Sage, Common Graphs (Graph Generators).
Eric Weisstein's World of Mathematics, Bipartite Graph.
Eric Weisstein's World of Mathematics, Bicolorable Graph.
Eric Weisstein's World of Mathematics, n-Colorable Graph.

Row sums of A297877.
The labeled version is A047864.
Equals A076278(n) + 1.
Cf. A005142 (connected).

A005142 = Cases[Import["https://oeis.org/A005142/b005142.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A005142] (* Jean-François Alcover, Mar 18 2020 *)

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

A033301 Number of 4-valent (or quartic) graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, 266, 1547, 10786, 88193, 805579, 8037796, 86223660, 985883873, 11946592242, 152808993767, 2056701139136, 29051369533596, 429669276147047, 6640178380127244, 107026751932268789, 1796103830404560857, 31334029441145918974, 567437704731717802783

Offset: 0

Ronald C. Read

Because the triangle A051031 is symmetric, a(n) is also the number of (n-5)-regular graphs on n vertices. - Jason Kimberley, Sep 22 2009

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

M. Meringer, Tables of Regular Graphs
M. Meringer, Erzeugung Regulärer Graphen, Diploma thesis, University of Bayreuth, January 1996. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010]
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Quartic Graph

4-regular simple graphs: A006820 (connected), A033483 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] (* Jean-François Alcover, Nov 26 2019, updated Mar 17 2020 *)

Euler transform of A006820. - Martin Fuller, Dec 04 2006

a(16) from Axel Kohnert (kohnert(AT)uni-bayreuth.de), Jul 24 2003
a(17)-a(19) from Jason Kimberley, Sep 12 2009
a(20)-a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
a(22) from Jason Kimberley, Oct 15 2011
a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 08 2020

A144958 Number of unlabeled acyclic graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 17, 39, 77, 176, 381, 891, 2057, 4941, 11915, 29391, 73058, 184236, 468330, 1202349, 3108760, 8097518, 21218776, 55925742, 148146312, 394300662, 1053929982, 2828250002, 7617271738, 20584886435, 55802753243

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of forests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A005195(n-1) counts the forests with one or more isolated nodes.

The labeled version is A105784. The connected case is A000055. This is the covering case of A005195. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of non-isomorphic representatives of the a(0) = 1 through a(5) = 4 forests:
  {}    .    {12}    {13,23}    {12,34}       {12,35,45}
                                {13,24,34}    {13,24,35,45}
                                {14,24,34}    {14,25,35,45}
                                              {15,25,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The connected case is A000055.
This is the covering case of A005195, labeled A001858.
The labeled version is A105784.
For triangles instead of cycles we have A372169, non-covering A006785.
Unique cycle: A372191 (lab A372195), non-covering A236570 (lab A372193).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A053530, A054548, A137917, A144959, A372174.

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}]]];
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}],And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Union[Union[brute/@Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[cyc[#]]==0&]]]],{n,0,5}] (* Gus Wiseman, Apr 29 2024 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), v=EulerT(Vec(t - t^2/2 + subst(t,x,x^2)/2))); concat([1,0], vector(#v-1, i, v[i+1]-v[i]))} \\ Andrew Howroyd, Aug 01 2024

a(n) = A005195(n) - A005195(n-1).

Name changed and 1 prepended by Gus Wiseman, Apr 29 2024.

A033483 Number of disconnected 4-valent (or quartic) graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 8, 25, 88, 378, 2026, 13351, 104595, 930586, 9124662, 96699987, 1095469608, 13175272208, 167460699184, 2241578965849, 31510542635443, 464047929509794, 7143991172244290, 114749135506381940, 1919658575933845129, 33393712487076999918, 603152722419661386031

Offset: 0

Ronald C. Read

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

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Quartic Graph

4-regular simple graphs: A006820 (connected), this sequence (disconnected), A033301 (not necessarily connected). - Jason Kimberley, Jan 08 2011
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), this sequence (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
Disconnected 4-regular simple graphs with girth at least g: this sequence (g=3), A185244 (g=4), A185245 (g=5), A185246 (g=6).

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] - A006820 (* Jean-François Alcover, Jan 21 2020, updated Mar 17 2020 *)

a(n) = A033301(n) - A006820(n) = Euler_transformation(A006820) - A006820.

a(n) = A068933(n, 4). - Jason Kimberley, Sep 27 2009 and Jan 08 2011

Terms a(16)-a(18) from Martin Fuller, Dec 04 2006
Terms a(19)-a(26) from Jason Kimberley, Sep 27 2009 and Dec 30 2010
Terms a(27)-a(33), due to the extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

cp's OEIS Frontend

A123423 Duplicate of A005195.

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A144978 A005195(n) - A005195(n-2).

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A000055 Number of trees with n unlabeled nodes.

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A001858 Number of forests of trees on n labeled nodes.

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A140637 Number of unlabeled graphs of positive excess with n nodes.

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A005470 Number of unlabeled planar simple graphs with n nodes.

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A033995 Number of bipartite graphs with n nodes.

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