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A000084
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Number of series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon.
(Formerly M1207 N0466)
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31
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1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624, 14136, 43930, 137908, 437502, 1399068, 4507352, 14611576, 47633486, 156047204, 513477502, 1696305728, 5623993944, 18706733128, 62408176762, 208769240140, 700129713630, 2353386723912
(list;
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listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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 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, Sep 27 2008]
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REFERENCES
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Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). DOI: http://dx.doi.org/10.1119/1.19396 - from Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Mar 06 2010
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.
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, E-Print archive: http://arxiv.org/abs/1004.3346/ (21 April 2010). [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Apr 27 2010]
S. A. Khan, How Many Equivalent Resistances?, RESONANCE, May 2012; http://www.ias.ac.in/resonance/May2012/p468-475.pdf. - From N. J. A. Sloane, Oct 15 2012
S. A. Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - From N. J. A. Sloane, Oct 23 2012
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. Reprinted in Discrete Appl. Math., 54 (1994), 225-228.
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.
Takeaki Uno, Ryuhei Uehara and Shin-ichi Nakano, Bounding the Number of Reduced Trees, Cographs, and Series-Parallel Graphs by Compression, in WALCOM: ALGORITHMS AND COMPUTATION, Lecture Notes in Computer Science, 2012, Volume 7157/2012, 5-16, DOI: 10.1007/978-3-642-28076-4_4. - N. J. A. Sloane, Jul 07 2012
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LINKS
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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
Sameen Ahmed KHAN, Mathematica notebook for A048211 and A000084
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
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FORMULA
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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.
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EXAMPLE
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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
....................... \/ ..\_/
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MAPLE
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(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);
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MATHEMATICA
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m = 27; b[1] = 1; b[n_ /; n >= 2] = a[n]/2;
ex = Product[ 1/(1 - x^k)^b[k], {k, 1, m}] - 1 - Sum[ a[k]*x^k, {k, 1, m}];
coes = CoefficientList[Series[ex, {x, 0, m}], x];
eq[0] = Thread[coes == 0] // Rest;
Do[s[k] = Solve[eq[k - 1][[1]], a[k]] // First; eq[k] = eq[k - 1] /. s[k] // Rest, {k, 1, m}];
Array[a, m] /. Flatten @ Table[s[k], {k, 1, m}]
(* From Jean-François Alcover, Jun 20 2011 *)
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PROG
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(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 */
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CROSSREFS
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Apart from initial term, 2*A000669. Cf. A058351, A058352, A058353, A000311, A006351 (labeled version).
See also A058964, A058965.
A144962, A001572 [From Gary W. Adamson, Sep 27 2008]
A176500, A176502 [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Apr 27 2010]
Sequence in context: A049146 A000682 A001997 * A057734 A151516 A003104
Adjacent sequences: A000081 A000082 A000083 * A000085 A000086 A000087
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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