Search: id:A006351
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%I A006351 M1885
%S A006351 1,2,8,52,472,5504,78416,1320064,25637824,564275648,13879795712,
%T A006351 377332365568,11234698041088,363581406419456,12707452084972544,
%U A006351 477027941930515456,19142041172838025216,817675811320888020992
%N A006351 Number of series-parallel networks with n labeled edges. Also called 
               yoke-chains by Cayley and MacMahon.
%D A006351 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006351 W. Knoedel, Ueber Zerfaellungen, Monatsh. Math., 55 (1951), 20-27.
%D A006351 Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 
               4 (1972), 109-150.
%D A006351 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.
%D A006351 P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 
               601-602; reprinted in Coll. Papers I, pp. 617-619.
%D A006351 J. W. Moon, Some enumerative results on series-parallel networks, Annals 
               Discrete Math., 33 (1987), 199-226 (the e.g.f. U(x)).
%D A006351 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               142.
%D A006351 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.40(a), S(x).
%H A006351 <a href="Sindx_Res.html#revert">Index entries for reversions of series</
               a>
%H A006351 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A006351 <a href="Sindx_Mo.html#Moon87">Index entries for sequences mentioned 
               in Moon (1987)</a>
%H A006351 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Series-parallel networks</
               a>
%F A006351 For n >= 2, A006351(n) = 2*A000311(n) = A005640(n)/2^n.
%F A006351 E.g.f. is reversion of 2*ln(1+x)-x.
%F A006351 Also exponential transform of A000311, define b by 1+sum b_n x^n / n! 
               = exp ( 1 + sum a_n x^n /n!).
%p A006351 read transforms; t1 := 2*ln(1+x)-x; t2 := series(t1,x,10); t3 := seriestoseries(t2,
               'revogf'); t4 := SERIESTOLISTMULT(%);
%p A006351 # 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);
%Y A006351 Cf. A000311, A000084 (for unlabeled case).
%Y A006351 Sequence in context: A125787 A007832 A111088 this_sequence A089467 A103239 
               A132228
%Y A006351 Adjacent sequences: A006348 A006349 A006350 this_sequence A006352 A006353 
               A006354
%K A006351 nonn,easy,nice
%O A006351 1,2
%A A006351 N. J. A. Sloane (njas(AT)research.att.com).

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