OFFSET
1,3
COMMENTS
Also the number of unlabeled connected cographs on n nodes. - N. J. A. Sloane and Eric W. Weisstein, Oct 21 2003
A cograph is a simple graph which contains no path of length 3 as an induced subgraph. - Michael Somos, Apr 19 2014
Also called "hierarchies" by Genitrini (2016). - N. J. A. Sloane, Mar 24 2017
REFERENCES
N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 43.
A. Brandstaedt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, SIAM Publications, 1999. (For definition of cograph)
A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 3, p. 246.
D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, p. 589, Answers to Exercises Section 2.3.4.4 5.
L. F. Meyers, Corrections and additions to Tree Representations in Linguistics. Report 3, 1966, p. 138. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
L. F. Meyers and W. S.-Y. Wang, Tree Representations in Linguistics. Report 3, 1963, pp. 107-108. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93 (the numbers called a_n in this paper). Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
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).
LINKS
N. J. A. Sloane, First 1001 terms of A000669
Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019-2021.
Florian Beck, Towards Learning Deep Rules, Ph. D. Thesis, J. Kepler Univ. (Linz, Austria, 2025). See p. 36.
Florian Beck, Johannes Fürnkranz, and Van Quoc Phuong Huynh, On the Potential of Deep Symbolic Models for Classification Problems, Int'l Conf. Discovery Sci., Lect. Notes Comp. Sci. (LNAI 2025) Vol. 16090.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. MR0891613 (89a:05009). See pp. 155, 162, 165, 172. - N. J. A. Sloane, Apr 18 2014
Peter J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian Zhong, Obstructions for three-coloring graphs without induced paths on six vertices, arXiv preprint arXiv:1504.06979 [math.CO], 2015-2018.
Audace Amen Vioutou Dossou-Olory, and Stephan Wagner, Inducibility of Topological Trees, arXiv:1802.06696 [math.CO], 2018.
Audace Amen Vioutou Dossou-Olory, The topological trees with extreme Matula numbers, arXiv:1806.03995 [math.CO], 2018-2020.
Steven R. Finch, Series-parallel networks.
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
Philippe Flajolet, A Problem in Statistical Classification Theory.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 72.
Daniel L. Geisler, Combinatorics of Iterated Functions.
Antoine Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], 2016, p. 9.
Olivier Golinelli, Asymptotic behavior of two-terminal series-parallel networks, arXiv:cond-mat/9707023 [cond-mat.stat-mech], 1997.
JiSun Huh and Seonjeong Park, Toric varieties of Schröder type, arXiv:2204.00214 [math.AG], 2022. (Table 1)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 44 [broken link].
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - N. J. A. Sloane, Dec 22 2012
Medet Jumadildayev, Enumeration of multipartite series-reduced trees, arXiv:2512.17216 [math.CO], 2025. See p. 19.
P. A. MacMahon, Yoke-chains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
Arnau Mir, Francesc Rossello, and Lucia Rotger, Sound Colless-like balance indices for multifurcating trees, arXiv:1805.01329 [q-bio.PE], 2018.
Vladimir Modrak and David Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299.
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
Vlady Ravelomanana and Loÿs Thimonier, Asymptotic enumeration of cographs, Electronic Notes in Discrete Mathematics, Volume 7, April 2001, pp. 58-61, Theorem 4.
John Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16. (page 6)
John Riordan, Letter to N. J. A. Sloane, Sep. 1970
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970
Wei Wang and Ximei Huang, Almost all cographs have a cospectral mate, arXiv:2507.16730 [math.CO], 2025. See pp. 6, 8.
Eric Weisstein's World of Mathematics, Series-Parallel Graph.
FORMULA
Product_{k>0} 1/(1-x^k)^a_k = 1+x+2*Sum_{k>1} a_k*x^k.
a(n) ~ c * d^n / n^(3/2), where d = 3.560839309538943329526129172709667..., c = 0.20638144460078903185013578707202765... [Ravelomanana and Thimonier, 2001]. - Vaclav Kotesovec, Aug 25 2014 [d is A395437. - Jianing Song, May 30 2026]
Consider a nontrivial partition p of n. For each size s of a part occurring in p, compute binomial(a(s)+m-1, m) where m is the multiplicity of s. Take the product of this expression over all s. Take the sum of this new expression over all p to obtain a(n). - Thomas Anton, Nov 22 2018
EXAMPLE
G.f. = x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 33*x^6 + 90*x^7 + 261*x^8 + ...
a(4)=5 with the following series-reduced planted trees: (oooo), (oo(oo)), (o(ooo)), (o(o(oo))), ((oo)(oo)). - Michael Somos, Jul 25 2003
MAPLE
Method 1: a := [1, 1]; for n from 3 to 30 do L := series( mul( (1-x^k)^(-a[k]), k=1..n-1)/(1-x^n)^b, x, n+1); t1 := coeff(L, x, n); R := series( 1+2*add(a[k]*x^k, k=1..n-1)+2*b*x^n, x, n+1); t2 := coeff(R, x, n); t3 := solve(t1-t2, b); a := [op(a), t3]; od: A000669 := n-> a[n];
Method 2, more efficient: with(numtheory): M := 1001; a := array(0..M); p := array(0..M); a[1] := 1; a[2] := 1; a[3] := 2; p[1] := 1; p[2] := 3; p[3] := 7;
Method 2, cont.: for m from 4 to M do t1 := divisors(m); t3 := 0; for d in t1 minus {m} do t3 := t3+d*a[d]; od: t4 := p[m-1]+2*add(p[k]*a[m-k], k=1..m-2)+t3; a[m] := t4/m; p[m] := t3+t4; od: # A000669 := n-> a[n]; A058757 := n->p[n];
# Method 3:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(a(i)+j-1, j)*
b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> `if`(n<2, n, b(n, n-1)):
seq(a(n), n=1..40); # Alois P. Heinz, Jan 28 2016
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i]+j-1, j]* b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<2, n, b[n, n-1]];
Array[a, 40] (* Jean-François Alcover, Jan 08 2021, after Alois P. Heinz *)
PROG
(PARI) {a(n) = my(A, X); if( n<2, n>0, X = x + x * O(x^n); A = 1 / (1 - X); for(k=2, n, A /= (1 - X^k)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Jul 25 2003 */
(SageMath)
from collections import Counter
def A000669_list(n):
list = [1] + [0] * (n - 1)
for i in range(1, n):
for p in Partitions(i + 1, min_length=2):
m = Counter(p)
list[i] += prod(binomial(list[s - 1] + m[s] - 1, m[s]) for s in m)
return list
print(A000669_list(20)) # M. Eren Kesim, Jun 21 2021
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Sequence cross reference fixed by Sean A. Irvine, Sep 15 2009
STATUS
approved