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Search: id:A006351
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| A006351 |
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Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon.
(Formerly M1885)
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10
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| 1, 2, 8, 52, 472, 5504, 78416, 1320064, 25637824, 564275648, 13879795712, 377332365568, 11234698041088, 363581406419456, 12707452084972544, 477027941930515456, 19142041172838025216, 817675811320888020992
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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W. Knoedel, Ueber Zerfaellungen, Monatsh. Math., 55 (1951), 20-27.
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 (the e.g.f. U(x)).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.40(a), S(x).
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LINKS
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Index entries for reversions of series
N. J. A. Sloane, Transforms
Index entries for sequences mentioned in Moon (1987)
S. R. Finch, Series-parallel networks
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FORMULA
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For n >= 2, A006351(n) = 2*A000311(n) = A005640(n)/2^n.
E.g.f. is reversion of 2*ln(1+x)-x.
Also exponential transform of A000311, define b by 1+sum b_n x^n / n! = exp ( 1 + sum a_n x^n /n!).
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MAPLE
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read transforms; t1 := 2*ln(1+x)-x; t2 := series(t1, x, 10); t3 := seriestoseries(t2, 'revogf'); t4 := SERIESTOLISTMULT(%);
# N denotes all series-parallel networks, S = series networks, P = parallel networks; spec := [ N, {N=Union(Z, S, P), S=Set(Union(Z, P), card>=2), P=Set(Union(Z, S), card>=2)}, labeled ]: A006351 := n->combstruct[count](spec, size=n);
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CROSSREFS
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Cf. A000311, A000084 (for unlabeled case).
Sequence in context: A125787 A007832 A111088 this_sequence A089467 A103239 A132228
Adjacent sequences: A006348 A006349 A006350 this_sequence A006352 A006353 A006354
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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