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 A004123 Number of generalized weak orders on n points. (Formerly M1975) 32
 1, 2, 10, 74, 730, 9002, 133210, 2299754, 45375130, 1007179562, 24840104410, 673895590634, 19944372341530, 639455369290922, 22079273878443610, 816812844197444714, 32232133532123179930, 1351401783010933015082 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003 From Peter Bala, Jul 08 2022: (Start) Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, ...] with an apparent period of 6 = phi(7) starting at a(2). Cf. A000670. More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End) REFERENCES L Santocanale, F Wehrung, G Grätzer, F Wehrung, Generalizations of the Permutohedron, in Grätzer G., Wehrung F. (eds) Lattice Theory: Special Topics and Applications. Birkhäuser, Cham, pp. 287-397; DOI https://doi.org/10.1007/978-3-319-44236-5_8 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 1..100 Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017. Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018. P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution. arXiv:quant-ph/0303030, 2003. C. G. Bower, Transforms D. Foata and C. Krattenthaler, Graphical Major Indices, II, Seminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995. D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994. Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020. Carl G. Wagner, Enumeration of generalized weak orders, Arch. Math. (Basel) 39 (1982), no. 2, 147-152. C. G. Wagner, Enumeration of generalized weak orders, Preprint, 1980. [Annotated scanned copy] C. G. Wagner and N. J. A. Sloane, Correspondence, 1980 FORMULA E.g.f.(for shifted sequence with offset 0): 1/(3-2*exp(x)). a(n) = 2^n*A(n,3/2); A(n,x) the Eulerian polynomials. - Peter Luschny, Aug 03 2010 O.g.f.: Sum_{n>=0} 2^n*n!*x^(n+1)/Product_{k=0..n} (1-k*x). - Paul D. Hanna, Jul 20 2011 a(n) = Sum_{k>=0} k^n*(2/3)^k/3. a(n) = Sum_{k=0..n} Stirling2(n, k)*(2^k)*k!. Stirling transform of A000165. - Karol A. Penson, Jan 25 2002 "AIJ" (ordered, indistinct, labeled) transform of 2, 2, 2, 2, ... Recurrence: a(n) = 2*Sum_{k=1..n} binomial(n, k)*a(n-k), a(0)=1. - Vladeta Jovovic, Mar 27 2003 G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)/(1-x-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 15 2013 a(n) ~ (n-1)!/(3*(log(3/2))^n). - Vaclav Kotesovec, Aug 07 2013 a(n) = log(3/2)*Integral_{x>=0} floor(x)^n * (3/2)^(-x) dx. - Peter Bala, Feb 14 2015 E.g.f.: (x - log(3 - 2*exp(x)))/3. - Ilya Gutkovskiy, May 31 2018 Conjectural o.g.f. as a continued fraction of Stieltjes type:  1/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 6*x/(1 - ... - 2*n*x/(1 - 3*n*x/(1 - ...))))))). - Peter Bala, Jul 08 2022 MATHEMATICA a[n_] := (1/3)*PolyLog[-n + 1, 2/3]; a[1]=1; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Jun 11 2012, after 3rd formula *) CoefficientList[Series[1/(3-2*Exp[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Aug 07 2013 *) PROG (PARI) {a(n)=polcoeff(sum(m=0, n, 2^m*m!*x^(m+1)/prod(k=1, m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */ (Sage) A004123 = lambda n: sum(stirling_number2(n-1, k)*(2^k)*factorial(k) for k in (0..n-1)) [A004123(n) for n in (1..18)] # Peter Luschny, Jan 18 2016 CROSSREFS Cf. A004121, A004122, A000165, A000670, A032033. Second row of array A094416 (generalized ordered Bell numbers). Equals 2 * A050351(n) for n>0. Sequence in context: A301932 A289313 A092881 * A086352 A005365 A191812 Adjacent sequences:  A004120 A004121 A004122 * A004124 A004125 A004126 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS More terms from Christian G. Bower STATUS approved

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Last modified August 14 05:20 EDT 2022. Contains 356110 sequences. (Running on oeis4.)