1,3

Also the number of unlabeled connected cographs on n nodes. - N. J. A. Sloane (njas(AT)research.att.com) and Eric W Weisstein, Oct 21, 2003.

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

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

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)

P. 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.

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.

P. A. MacMahon, Yoke-trains 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.

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. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.

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, First 1001 terms of A000669

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

S. R. Finch, Series-parallel networks

Philippe Flajolet, A Problem in Statistical Classification Theory

Daniel L. Geisler, Combinatorics of Iterated Functions

O. Golinelli, Asymptotic behavior of two-terminal series-parallel networks.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 44

Eric Weisstein's World of Mathematics, Series-Parallel Network

Index entries for sequences related to rooted trees

Index entries for sequences mentioned in Moon (1987)

Index entries for sequences related to trees

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 72

Product_{k>0} 1/(1-x^k)^a_k = 1+x+2*Sum_{k>1} a_k*x^k.

a(4)=5 with the following series-reduced planted trees: (oooo), (oo(oo)), (o(ooo)), (o(o(oo))), ((oo)(oo)).

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];

(PARI) a(n)=local(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)

Equals (1/2)*A000084 for n >= 2. Cf. A000055, A000311, A001678, A007827.

Cf. A000311, labeled hierarchies on n points.

Sequence in context: A084075 A000560 A032124 this_sequence A004114 A076864 A032292

Adjacent sequences: A000666 A000667 A000668 this_sequence A000670 A000671 A000672

nonn,nice,easy

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

Sequence cross reference fixed by Sean A. Irvine (sairvin(AT)xtra.co.nz), Sep 15 2009