1,2

This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel.

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

Also the number of P_4-free graphs on n nodes. - Gordon Royle, Jul 04 2008

Equals row sums of triangle A144962 and the INVERT transform of A001572. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 27 2008]

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.

S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.

D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, p. 589, Answers to Exercises Section 2.3.4.4 5.

Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.

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.

P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.

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

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.

J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. 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).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.40, notes on p. 133.

N. J. A. Sloane, Table of n, a(n) for n=1..1001

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

S. R. Finch, Series-parallel networks

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

S. Hougardy, Home Page

N. J. A. Sloane, First 1001 terms of A000669 and A000084

Eric Weisstein's World of Mathematics, Cograph

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

Let b(1)=1, b(n)=a(n)/2 for n >= 2. Then sequence satisfies Product_{k=1..inf} 1/(1-x^k)^b(k) = 1 + Sum_{k=1..inf} a(k)*x^k.

a(n) ~ C d^n/n^(3/2) where C = 0.4126..., d = 3.560839309538943329526... is a root of Prod_{ n >= 1} (1-1/x^n)^(-a(n)) = 2. - Riordan, Shannon, Moon, Rains, Sloane

Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and one generator A. The number of elements with n occurrences of the generator is a(n). - Michael Somos Oct 11 2006. Examples: n=1: A. n=2: A+A, A*A. n=3: A+A+A, A+(A*A), A*(A+A), A*A*A.

The series-parallel networks with 1, 2 and 3 edges are:

1 edge: o-o

2 edges: o-o-o o=o

....................... /\

3 edges: o-o-o-o o-o=o o--o o-o-o

....................... \/ ..\_/

(continue from A000669) A000084 := n-> if n=1 then 1 else 2*A000669(n); fi;

# N denotes all series-parallel networks, S = series networks, P = parallel networks; spec84 := [ N, {N=Union(Z, S, P), S=Set(Union(Z, P), card>=2), P=Set(Union(Z, S), card>=2)} ]: A000084 := n->combstruct[count](spec84, size=n);

(PARI) {a(n)=local(A); if(n<1, 0, A=1/(1-x+x*O(x^n)); for(k=2, n, A/=(1-x^k+x*O(x^n))^polcoeff(A, k)); polcoeff(A, n))} /* Michael Somos Oct 11 2006 */

Apart from initial term, 2*A000669. Cf. A058351, A058352, A058353, A000311, A006351 (labeled version).

See also A058964, A058965.

A144962, A001572 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 27 2008]

Sequence in context: A049146 A000682 A001997 this_sequence A057734 A003104 A151516

Adjacent sequences: A000081 A000082 A000083 this_sequence A000085 A000086 A000087

nonn,nice,easy

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