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.

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]

Adjacent sequences: A000081 A000082 A000083 this_sequence A000085 A000086 A000087

Sequence in context: A049146 A000682 A001997 this_sequence A057734 A003104 A151516

nonn,nice,easy

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