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 A006905 Number of transitive relations on n labeled nodes. (Formerly M2065) 9
 1, 2, 13, 171, 3994, 154303, 9415189, 878222530, 122207703623, 24890747921947, 7307450299510288, 3053521546333103057, 1797003559223770324237, 1476062693867019126073312, 1679239558149570229156802997, 2628225174143857306623695576671, 5626175867513779058707006016592954, 16388270713364863943791979866838296851, 64662720846908542794678859718227127212465 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES D. Ford and J. McKay, personal communication, 1991. Klaska (1997), Transitivity and Partial Order, Mathematica Bohemica, 122 (1), 75-82. Based on a correspondence between transitive relations and partial orders, the author obtains a formula and calculates the first 14 terms - Jeff McSweeney (jmcsween(AT)mtsu.edu), May 13, 2000 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS S. R. Finch, Transitive relations, topologies and partial orders S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author] G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2. FORMULA E.g.f.: A(x + exp(x) - 1) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014 MATHEMATICA P = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]]; a[n_] := Sum[P[[k+1]] Sum[Binomial[n, s] StirlingS2[n-s, k-s], {s, 0, k}], {k, 0, n}]; a /@ Range[0, 18] (* Jean-François Alcover, Dec 29 2019, after Charles R Greathouse IV *) PROG (PARI) \\ P = [1, 1, 3, 19, ...] is A001035, starting from 0. a(n)=sum(k=0, n, P[k+1]*sum(s=0, k, binomial(n, s)*stirling(n-s, k-s, 2))) \\ Charles R Greathouse IV, Sep 05 2011 CROSSREFS Cf. A000595, A001173. See A091073 for unlabeled case. Sequence in context: A078363 A143851 A088316 * A119400 A182314 A268988 Adjacent sequences:  A006902 A006903 A006904 * A006906 A006907 A006908 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003 a(15)-a(16) from Charles R Greathouse IV, Aug 30 2006 a(17)-a(18) from Charles R Greathouse IV, Sep 05 2011 STATUS approved

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Last modified September 18 07:17 EDT 2020. Contains 337166 sequences. (Running on oeis4.)