A054924 -id:A054924 - cp's OEIS frontend

A000055 Number of trees with n unlabeled nodes.

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

A001349 Number of simple connected graphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476, 50335907869219, 29003487462848061, 31397381142761241960, 63969560113225176176277, 245871831682084026519528568, 1787331725248899088890200576580, 24636021429399867655322650759681644

Offset: 0

N. J. A. Sloane

The singleton graph K_1 is considered connected even though it is conventionally taken to have vertex connectivity 0. - Eric W. Weisstein, Jul 21 2020

Inverse Euler transform of A000088 but with a(0) omitted so that Sum_{k>=0} A000088(n) * x^n = Product_{k>0} (1 - x^k)^-a(k). It is debatable if there is a connected graph with 0 nodes and so a(0)=0 or better start from a(1)=1. - Michael Somos, Jun 01 2013. [As Harary once remarked in a famous paper ("Is the null-graph a pointless concept?"), the empty graph has every property, which is why a(0)=1. - N. J. A. Sloane, Apr 08 2014]

			G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 21*x^5 + 112*x^6 + 853*x^7 + ....

P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.
F. Harary and R. C. Read, Is the null-graph a pointless concept?, pp. 37-44 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, page 48, c(x). Also page 242.
Lupanov, O. B. Asymptotic estimates of the number of graphs with n edges. (Russian) Dokl. Akad. Nauk SSSR 126 1959 498--500. MR0109796 (22 #681).
Lupanov, O. B. "On asymptotic estimates of the number of graphs and networks with n edges." Problems of Cybernetics [in Russian], Moscow 4 (1960): 5-21.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.
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).
Robin J. Wilson, Introduction to Graph Theory, Academic Press, 1972. (But see A126060!)

N. J. A. Sloane, Table of n, a(n) for n = 0..75 [Computed using Keith Briggs's values for A000088]
Michal Adamaszek, Small flag complexes with torsion, arXiv:1208.3892 [math.CO], 2012.
C. O. Aguilar and B. Gharesifard, Graph Controllability Classes for the Laplacian Leader-Follower Dynamics, 2014. See Table 1.
Jonathan Baker, Kevin N. Vander Meulen, and Adam Van Tuyl, Shedding vertices of vertex decomposable well-covered graphs, Discrete Mathematics (2018) Vol. 341, Issue 12, 3355-3369. Also arXiv:1606.04447 [math.NT], 2016.
Johannes Bausch and Felix Leditzky, Error Thresholds for Arbitrary Pauli Noise, arXiv:1910.00471 [quant-ph], 2019.
Gunnar Brinkmann, Kris Coolsaet, Jan Goedgebeur and Hadrien Melot, House of Graphs: a database of interesting graphs, arXiv:1204.3549 [math.CO], 2012.
P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp. [Annotated scanned copy]
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
CombOS - Combinatorial Object Server, Generate graphs
Matt DeVos, Adam Dyck, Jonathan Jedwab, and Samuel Simon, Which graphs occur as gamma-graphs?, arXiv:1810.01583 [math.CO], 2018.
J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271. (Annotated scanned copy of 3 pages)
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
E. Friedman, Illustration of small graphs
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463.
Yoshiaki Horiike and Yuki Kawaguchi, A comprehensive exploration of interaction networks reveals a connection between entanglement and network structure, arXiv:2505.11466 [quant-ph], 2025. See p. 9.
Avraham Itzhakov and Michael Codish, Incremental Symmetry Breaking Constraints for Graph Search Problems, Ben-Gurion University of the Negev (Beer-Sheva, Israel, 2019).
Markus Kirchweger and Stefan Szeider, SAT Modulo Symmetries for Graph Generation and Enumeration, ACM Trans. Comput. Logic (2024). See p. 27.
X. Li, D. S. Stones, H. Wang, H. Deng, X. Liu and G. Wang, NetMODE: Network Motif Detection without Nauty, PLoS ONE 7(12): e50093. - From _N. J. A. Sloane_, Feb 02 2013
Steffen Lauritzen, Alessandro Rinaldo, and Kayvan Sadeghi, On Exchangeability in Network Models, arXiv:1709.03885 [math.ST], 2017.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
B. D. McKay, Simple Graphs
A. Milicevic and N. Trinajstic, Combinatorial Enumeration in Chemistry, Chem. Modell., Vol. 4, (2006), pp. 405-469.
Marius Möller, Laura Hindersin, and Arne Traulsen, Exploring and mapping the universe of evolutionary graphs, arXiv:1810.12807 [q-bio.PE], 2018.
Marius Möller, Laura Hindersin, and Arne Traulsen, Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time, Communications Biology 2 (2019), Article number: 137.
L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8
M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
Gordon Royle, Small graphs
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M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967.
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.)
D. Stolee, Isomorph-free generation of 2-connected graphs with applications, arXiv:1104.5261 [math.CO], 2011.
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Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, k-Connected Graph
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Index entries for "core" sequences

Cf. A000088, A002218, A006290, A000719, A201922 (Multiset transform).

Row sums of A054924.

# To produce all connected graphs on 4 nodes, for example (from N. J. A. Sloane, Oct 07 2013):
with(GraphTheory):
L:=[NonIsomorphicGraphs](4,output=graphs,outputform=adjacency, restrictto=connected):

<<"Combinatorica`"; max = 19; A000088 = Table[ NumberOfGraphs[n], {n, 0, max}]; f[x_] = 1 - Product[ 1/(1 - x^k)^a[k], {k, 1, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[ f[x], {x, 0, max}], x]; sol = First[ Solve[ Thread[ Rest[ coes + A000088 ] == 0]]]; Table[ a[n], {n, 0, max}] /. sol (* Jean-François Alcover, Nov 24 2011 *)
terms = 20;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Join[{1}, EULERi[Array[a88, terms]]] (* Jean-François Alcover, Jul 28 2018, after Andrew Howroyd *)

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A001349(n):
    if n == 0: return 1
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 02-03 2024

property=lambda G: G.is_connected()
def a(n):
    return len([1 for G in graphs(n) if property(G)])
# Ralf Stephan, May 30 2014

For asymptotics see Lupanov 1959, 1960, also Turner and Kautz, p. 18. - N. J. A. Sloane, Apr 08 2014

More terms from Ronald C. Read

A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1, 1, 1, 2, 5, 9, 15, 21, 24, 24, 21, 15, 9, 5, 2, 1, 1, 1, 1, 2, 5, 10, 21, 41, 65, 97, 131, 148, 148, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1, 1, 1, 2, 5, 11, 24, 56, 115, 221, 402, 663, 980, 1312, 1557, 1646, 1557

Offset: 1

N. J. A. Sloane, Mar 15 1996

T(n,k)=1 for n>=2 with k=0, k=1, k=n*(n-1)/2-1 and k=n*(n-1)/2 (therefore the quadruple {1,1,1,1} marks the transition to the next sublist for a given number of vertices (n>2)). [Edited by Peter Munn, Mar 20 2021]

			Triangle begins:
1,
1,1,
1,1,1,1,
1,1,2,3,2,1,1, [graphs with 4 nodes and from 0 to 6 edges]
1,1,2,4,6,6,6,4,2,1,1,
1,1,2,5,9,15,21,24,24,21,15,9,5,2,1,1,
1,1,2,5,10,21,41,65,97,131,148,148,131,97,65,41,21,10,5,2,1,1,
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 264.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 519.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 240.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. W. Robinson, Rows 1 to 20 of triangle, flattened
Leonid Bedratyuk, A new formula for the generating function of the numbers of simple graphs, arXiv:1512.06355 [math.CO], 2015.
FindStat - Combinatorial Statistic Finder, The number of edges of a graph.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 65.
Sriram V. Pemmaraju, Combinatorica 2.0
Marko R. Riedel, Number of distinct connected digraphs
Gordon Royle, Small graphs
S. S. Skiena, Generating graphs
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Overview of the 12 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 1
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 2
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 3
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 4
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 5
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 6
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 7
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 8
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 9
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 10
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 11
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 12
James Turner and William H. Kautz, 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.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Simple Graph
A. E. Yurtsun, Principles of enumeration of the number of graphs, Ukrainian Mathematical Journal, January-February, 1967, Volume 19, Issue 1, pp 123-125, DOI 10.1007/BF01085184.

Row sums give A000088.
Cf. A046742, A046751, A000717, A001432, A001431, A001430, A001433, A001434, A048179, A048180 etc.
Cf. also A039735, A002905, A054924 (connected), A084546 (labeled graphs).
Row lengths: A000124; number of connected graphs for given number of vertices: A001349; number of graphs for given number of edges: A000664.
Cf. also A000055.

seq(seq(GraphTheory:-NonIsomorphicGraphs(v,e),e=0..v*(v-1)/2),v=1..9); # Robert Israel, Dec 22 2015

<< Combinatorica`; Table[CoefficientList[GraphPolynomial[n, x], x], {n, 8}] // Flatten (* Eric W. Weisstein, Mar 20 2013 *)
<< Combinatorica`; Table[NumberOfGraphs[v, e], {v, 8}, {e, 0, Binomial[v, 2]}] // Flatten (* Eric W. Weisstein, May 17 2017 *)
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/ g]^g,{j, 1, i-1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[ c-1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, 1 + x^#&], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x]&;
Array[row, 8] // Flatten (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, A=0) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i+A)); s/n!}
{ for(n=1, 7, print(Vecrev(G(n)))) } \\ Andrew Howroyd, Oct 22 2019, updated  Jan 09 2024

def T(n,k):
    return len(list(graphs(n, size=k)))
# Ralf Stephan, May 30 2014

O.g.f. for n-th row: 1/n! Sum_g det(1-g z^2)/det(1-g z) where g runs through the natural matrix representation of the pair group A^2_n (for A^2_n see F. Harary and E. M. Palmer, Graphical Enumeration, page 83). - Leonid Bedratyuk, Sep 23 2014

Additional comments from Arne Ring (arne.ring(AT)epost.de), Oct 03 2002

Text belonging in a different sequence deleted by Peter Munn, Mar 20 2021

A001429 Number of n-node connected unicyclic graphs.

Original entry on oeis.org

1, 2, 5, 13, 33, 89, 240, 657, 1806, 5026, 13999, 39260, 110381, 311465, 880840, 2497405, 7093751, 20187313, 57537552, 164235501, 469406091, 1343268050, 3848223585, 11035981711, 31679671920, 91021354454, 261741776369, 753265624291, 2169441973139, 6252511838796

Offset: 3

N. J. A. Sloane

Also unlabeled connected simple graphs with n vertices and n edges. The labeled version is A057500. - Gus Wiseman, Feb 12 2024

			From _Gus Wiseman_, Feb 12 2024: (Start)
Representatives of the a(3) = 1 through a(6) = 13 simple graphs:
  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}  {12,13,14,15,16,23}
              {12,13,24,34}  {12,13,14,23,25}  {12,13,14,15,23,26}
                             {12,13,14,23,45}  {12,13,14,15,23,46}
                             {12,13,14,25,35}  {12,13,14,15,26,36}
                             {12,13,24,35,45}  {12,13,14,23,25,36}
                                               {12,13,14,23,25,46}
                                               {12,13,14,23,45,46}
                                               {12,13,14,23,45,56}
                                               {12,13,14,25,26,35}
                                               {12,13,14,25,35,46}
                                               {12,13,14,25,35,56}
                                               {12,13,14,25,36,56}
                                               {12,13,24,35,46,56}
(End)

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
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).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Washington G. Bomfim, A picture of the twenty one unicycles with 3,4,5 and 6 vertices
Audace A. V. Dossou-Olory, Graphs and unicyclic graphs with extremal connected subgraphs, arXiv:1812.02422 [math.CO], 2018.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
S. Karim, J. Sawada, Z. Alamgirz and S. M. Husnine, Generating bracelets with fixed content, Theoretical Computer Science, Volume 475, 4 March 2013, Pages 103-112.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Marko Riedel et al., Non-isomorphic, connected, unicyclic graphs, Math Stackexchange, November 2018. (Derivation of algorithm and Maple implementation using PET.)
Marko Riedel, Maple implementation using PET.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points, Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967. doi: 10.2172/4180737.
Eric Weisstein's World of Mathematics, Unicyclic Graph

For at most one cycle we have A005703, labeled A129271, complement A140637.
Next-to-main diagonal of A054924. Cf. A000055.
The labeled version is A057500, connected case of A137916.
This is the connected case of A137917 and A236570.
Row k = 1 of A137918.
The version with loops is A368983.
A001349 counts unlabeled connected graphs.
A001434 and A006649 count unlabeled graphs with # vertices = # edges.
A006129 counts covering graphs, unlabeled A002494.
Cf. A014068, A054548, A116508, A133686, A140638, A369143, A369201.

Needs["Combinatorica`"];
nn=30;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);rt=Table[a[i],{i,1,nn}];Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
(* Second program: *)
TreeGf[nn_] := Module[{A}, A = Table[1, {nn}]; For[n = 1, n <= nn 1, n++, A[[n + 1]] = 1/n * Sum[Sum[ d*A[[d]], {d, Divisors[k]}]*A[[n - k + 1]], {k, 1, n}]]; x A.x^Range[0, nn-1]];
seq[n_] := Module[{t, g}, If[n < 3, {}, t = TreeGf[n - 2]; g[e_] := Normal[t + O[x]^(Quotient[n, e]+1)] /. x -> x^e  + O[x]^(n+1); Sum[Sum[ EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[k], g[1]* g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2], {k, 3, n}]]/2 // Drop[CoefficientList[#, x], 3]&];
seq[32] (* Jean-François Alcover, Oct 05 2019, after Andrew Howroyd's PARI code *)

\\ TreeGf gives gf of A000081
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)={if(n<3, [], my(t=TreeGf(n-2)); my(g(e)=subst(t + O(x*x^(n\e)),x,x^e) + O(x*x^n)); Vec(sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2))} \\ Andrew Howroyd, May 05 2018

a(n) = A068051(n) - A027852(n) - A000081(n).

More terms from Ronald C. Read
a(27) corrected, more terms, formula from Christian G. Bower, Feb 12 2002
Edited by Charles R Greathouse IV, Oct 05 2009
Terms a(30) and beyond from Andrew Howroyd, May 05 2018

A002905 Number of connected graphs with n edges.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 30, 79, 227, 710, 2322, 8071, 29503, 112822, 450141, 1867871, 8037472, 35787667, 164551477, 779945969, 3804967442, 19079312775, 98211456209, 518397621443, 2802993986619, 15510781288250, 87765472487659, 507395402140211, 2994893000122118, 18035546081743772, 110741792670074054, 692894304050453139

Offset: 0

N. J. A. Sloane

			a(3) = 3 since the three connected graphs with three edges are a path, a triangle and a "Y".
The first difference between this sequence and A046091 is for n=9 edges where we see K_{3,3}, the well-known "utility graph".

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).

Max Alekseyev, Table of n, a(n) for n = 0..60
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
Nicolas Borie, The Hopf Algebra of graph invariants, arXiv preprint arXiv:1511.05843 [math.CO], 2015.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Mike Cummings and Adam Van Tuyl, The GeometricDecomposability package for Macaulay2, arXiv:2211.02471 [math.AC], 2022.
Anjan Dutta and Hichem Sahbi, Graph Kernels based on High Order Graphlet Parsing and Hashing, arXiv:1803.00425 [cs.CV], 2018.
Gordon Royle, Small graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 1 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Polynema.

Column sums of A054924 or equivalently row sums of A054923.
Cf. A000664, A046091 (for connected planar graphs), A275421 (multisets).
Apart from a(3), same as A003089.

A000664 = Cases[Import["https://oeis.org/A000664/b000664.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[Rest @ A000664]] (* Jean-François Alcover, May 10 2019, updated Mar 17 2020 *)

A000664 and this sequence are an Euler transform pair. - N. J. A. Sloane, Aug 30 2016

More terms from Vladeta Jovovic, Jan 12 2000
More terms from Gordon F. Royle, Jun 05 2003
a(25)-a(26) from Max Alekseyev, Sep 19 2009
a(27)-a(60) from Max Alekseyev, Sep 07 2016

A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 5, 6, 0, 0, 0, 1, 5, 13, 11, 0, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

Offset: 0

N. J. A. Sloane

The diagonal n = k+1 is A000055(n). - Jonathan Vos Post, Aug 10 2008

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 5   6;
  0, 0, 0, 1, 5, 13,  11;
  0, 0, 0, 0, 4, 19,  33,  23;
  0, 0, 0, 0, 2, 22,  67,  89,  47;
  0, 0, 0, 0, 1, 20, 107, 236, 240, 106;
  ... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 6 with 6 nodes). [Typo corrected by Anders Haglund, Jul 08 2008]

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 93, Table 4.2.2; p. 241, Table A2.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 57.
Gordon Royle, Small graphs
Gus Wiseman, Non-isomorphic representatives of the 12 connected graphs counted in row 5.

Main diagonal is A000055.
Subsequent diagonals give the number of connected unlabeled graphs with n nodes and n+k edges for k=0..2: A001429, A001435, A001436.
Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column), A343088 (labeled).
Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151.

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 23 2019

a(83)-a(89) corrected by Andrew Howroyd, Oct 24 2019

A062734 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 16, 15, 6, 1, 0, 0, 0, 0, 125, 222, 205, 120, 45, 10, 1, 0, 0, 0, 0, 0, 1296, 3660, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 0, 0, 16807, 68295, 156555, 258125, 331506, 343140, 290745, 202755, 116175, 54257, 20349

Offset: 1

Vladeta Jovovic, Jul 12 2001

T(n,n-1) = n^(n-2) counts free labeled trees A000272.

T(n,n) counts labeled connected unicyclic graphs A057500. - Geoffrey Critzer, Oct 07 2012

			Triangle starts:
[1],
[0, 1],
[0, 0, 3,  1],
[0, 0, 0, 16,  15,   6,   1],
[0, 0, 0,  0, 125, 222, 205, 120, 45, 10, 1],
...

Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519). - N. J. A. Sloane, Apr 06 2012
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.5.

Seiichi Manyama, Table of n, a(n) for n = 1..9919 (terms 1..75 from Alex Ermolaev, terms 76..175 from Alois P. Heinz)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; Table 58.

Cf. A001187 (row sums), A054924 (unlabeled case), A061540 (a subdiagonal).

See A123527 for another version (without leading zeros in each row).

nn=6;s=Sum[(1+y)^Binomial[n,2] x^n/n!,{n,0,nn}]; Range[0,nn]!CoefficientList[Series[Log[ s]+1,{x,0,nn}],{x,y}]//Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, k]]; (* Michael Somos, Aug 12 2017 *)

{T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( log( sum(m=0, n, (1 + y)^(m * (m-1)/2) * x^m/m!)), n), k))}; /* Michael Somos, Aug 12 2017 */

G.f.: Sum_{n>=1, k>=0} T(n, k) * x^n/n! * y^k = log(Sum_{n>=0} (1 + y)^binomial(n, 2) * x^n/n!). - Ralf Stephan, Jan 18 2005

A001434 Number of graphs with n nodes and n edges.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 21, 65, 221, 771, 2769, 10250, 39243, 154658, 628635, 2632420, 11353457, 50411413, 230341716, 1082481189, 5228952960, 25945377057, 132140242356, 690238318754, 3694876952577, 20252697246580, 113578669178222, 651178533855913, 3813856010041981

Offset: 0

N. J. A. Sloane

The labeled version is A116508. - Gus Wiseman, Feb 22 2024

			From _Gus Wiseman_, Feb 22 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 6 graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,24}
                                       {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..40 from Sean A. Irvine)
CombOS - Combinatorial Object Server, Generate graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967.

The connected case is A001429, labeled A057500.
The covering case is A006649, labeled A367863.
Diagonal n = k of A008406.
The labeled version is A116508.
The version with loops is A368598, connected A368983.
Allowing up to n edges gives A370315, labeled A369192.
A000088 counts unlabeled graphs, labeled A006125.
A001349 counts unlabeled connected graphs, labeled A001187.
A002494 counts unlabeled covering graphs, labeled A006129.
Cf. A000055, A005703, A014068, A054924, A137916, A137917, A236570, A369201.

(* first do *) Needs["Combinatorica`"] (* then *) Table[ NumberOfGraphs[n, n], {n, 24}] (* Robert G. Wilson v, Mar 22 2011 *)
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 /@ Subsets[Subsets[Range[n],{2}],{n}]]],{n,0,5}] (* Gus Wiseman, Feb 22 2024 *)

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A008406. - Andrew Howroyd, Feb 02 2024

More terms from Vladeta Jovovic, Jan 07 2000

a(0)=1 prepended by Andrew Howroyd, Feb 02 2024

A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 1, 6, 20, 41, 73, 110, 133, 139, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 1, 3, 15, 50, 124, 271, 515, 832, 1181, 1460, 1581, 1516, 1291, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Gus Wiseman, Feb 15 2024

			Triangle begins:
  1
  0
  0  1
  0  0  1  1
  0  0  1  2  2  1  1
  0  0  0  1  4  5  5  4  2  1  1
  0  0  0  1  3  9 15 20 22 20 14  9  5  2  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Column sums are A000664.
Row sums are A002494.
This is the covering case of A008406, labeled A084546.
The labeled version is A054548, row sums A006129, column sums A121251.
The connected case is A054924, row sums A001349, column sums A002905.
The labeled connected case is A062734, with loops A369195.
The connected case with loops is A283755, row sums A054921.
The labeled version w/ loops is A369199, row sums A322661, col sums A173219.
Cf. A000666, A006125, A006649, A054547, A066383, A322700.

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}],{k}],Union@@#==Range[n]&]]], {n,0,5},{k,0,Binomial[n,2]}]

\\ G(n) defined in A008406.
row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n,2)+1)}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

a(42) onwards from Andrew Howroyd, Feb 19 2024

A054925 a(n) = ceiling(n*(n-1)/4).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, 716, 743, 770, 798

Offset: 0

N. J. A. Sloane, May 24 2000

Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
Form the clockwise spiral starting 0,1,2,....; then A054925(n+1) interleaves 2 horizontal (A033951, A033991) and 2 vertical (A007742, A054552) branches. A bisection is A014848. - Paul Barry, Oct 08 2007
Consider the standard 4-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 2 along the positive y-axis, 3 along the positive z-axis, 4 along the positive t-axis, and then back round to the x-axis. This sequence gives the floor of the Euclidean distance to the origin after n steps. - Jon Perry, Apr 16 2013
Jon Perry's JavaScript code is explained by A238604. - Michael Somos, Mar 01 2014
Ceiling of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Ceiling of one-half of each triangular number. - Harvey P. Dale, Oct 03 2016
For n > 2, also the edge cover number of the (n-1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
Conjecture: For n>11, there always exists a prime number p such that a(n)Raul Prisacariu, Sep 01 2024
For n = 1 up to at least n = 13, also the lower matching number of the triangular honeycomb bishop graph. - Eric W. Weisstein, Dec 13 2024
Conjecturally, apart from the first term, the sequence terms are the exponents in the expansion of Sum_{n >= 0} q^(3*n) * (Product_{k >= 2*n+1} 1 - q^k) = 1 - q - q^2 + q^3 + q^5 - q^8 - q^11 + + - - .... Cf. A039825. - Peter Bala, Apr 13 2025

			a(6) = 8; ceiling(6*(6-1)/4) = ceiling(30/4) = 8.
G.f. = x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 18*x^9 + 23*x^10 + ...

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Craig Knecht, Binary fill patterns in a triangle refilled with natural numbers.
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A007742, A033951, A033991, A039825, A054552, A054924, A054925 + A011848 = C(n, 2).

Cf. A213172, A238604.

p=new Array(0,0,0,0);
for (a=0;a<100;a++) {
p[a%4]+=a;
document.write(Math.floor(Math.sqrt(p[0]*p[0]+p[1]*p[1]+p[2]*p[2]+p[3]*p[3]))+", ");
} /* Jon Perry, Apr 16 2013 */

[ Ceiling(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

I:=[0,0,1,2,3]; [n le 5 select I[n] else 3*Self(n-1)-4*Self(n-2)+4*Self(n-3)-3*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

seq(ceil(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Ceiling[(n^2 - n)/4], {n, 0, 20}] (* Wesley Ivan Hurt, Nov 01 2013 *)
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 0, 1, 2, 3}, 60] (* Vincenzo Librandi, Jul 14 2015 *)
Join[{0}, Ceiling[#/2] &/ @ Accumulate[Range[0, 60]]] (* Harvey P. Dale, Oct 03 2016 *)
Ceiling[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Dec 13 2024 *)
Table[Ceiling[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
Table[(1 + (n - 1) n - Cos[n Pi/2] - Sin[n Pi/2])/4, {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
CoefficientList[Series[x^2 (-1 + x - x^2)/((-1 + x)^3 (1 + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 13 2024 *)

{a(n) = ceil( n * (n-1)/4)}; /* Michael Somos, Feb 11 2004 */

[ceil(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - Michael Somos, Sep 02 2006
From Michael Somos, Feb 11 2004: (Start)
G.f.: x^2 * (x^2 - x + 1) / ((1 - x)^3 * (1 + x^2)) = x^2 * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)).
a(1-n) = a(n).
A011848(n) = a(-n). (End)
From Michael Somos, Mar 01 2014: (Start)
a(n + 4) = a(n) + 2*n + 3.
a(n+1) = floor( sqrt( A238604(n))). (End)
a(n) = A011848(n) + A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
Sum_{n>=2} 1/a(n) = 4 - Pi + 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(1/sqrt(2)+cosh(sqrt(7)*Pi/4))). - Amiram Eldar, Dec 23 2024

cp's OEIS Frontend

A000055 Number of trees with n unlabeled nodes.

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A001349 Number of simple connected graphs on n unlabeled nodes.

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A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).

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A001429 Number of n-node connected unicyclic graphs.

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A002905 Number of connected graphs with n edges.

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A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

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