1,3

Also the number of unlabeled connected cographs on n nodes. - njas and Eric W Weisstein, Oct 21, 2003.

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

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]; A058575 := 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

njas, John Riordan