%I M1975 #96 May 03 2024 01:16:45
%S 1,2,10,74,730,9002,133210,2299754,45375130,1007179562,24840104410,
%T 673895590634,19944372341530,639455369290922,22079273878443610,
%U 816812844197444714,32232133532123179930,1351401783010933015082
%N Number of generalized weak orders on n points.
%C Number of bipartitional relations on a set of cardinality n. - _Ralf Stephan_, Apr 27 2003
%C From _Peter Bala_, Jul 08 2022: (Start)
%C 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.
%C 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)
%D 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A004123/b004123.txt">Table of n, a(n) for n = 1..100</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1702.04007">Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays</a>, arXiv:1702.04007 [math.CO], 2017.
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0303030">Dobinski-type relations and the log-normal distribution</a>. arXiv:quant-ph/0303030, 2003.
%H C. G. Bower, <a href="/transforms2.html">Transforms</a>
%H D. Foata and C. Krattenthaler, <a href="http://www.mat.univie.ac.at/~kratt/artikel/graphmaj.html">Graphical Major Indices, II</a>, Seminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
%H D. Foata and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9406220">The Graphical Major Index</a>, arXiv:math/9406220 [math.CO], 1994.
%H Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%H Carl G. Wagner, <a href="http://dx.doi.org/10.1007/BF01899195">Enumeration of generalized weak orders</a>, Arch. Math. (Basel) 39 (1982), no. 2, 147-152.
%H C. G. Wagner, <a href="/A004121/a004121_1.pdf">Enumeration of generalized weak orders</a>, Preprint, 1980. [Annotated scanned copy]
%H C. G. Wagner and N. J. A. Sloane, <a href="/A004121/a004121.pdf">Correspondence, 1980</a>
%F E.g.f. for sequence with offset 0: 1/(3-2*exp(x)).
%F a(n) = 2^n*A(n,3/2); A(n,x) the Eulerian polynomials. - _Peter Luschny_, Aug 03 2010
%F 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
%F a(n) = Sum_{k>=0} k^n*(2/3)^k/3.
%F a(n) = Sum_{k=0..n} Stirling2(n, k)*(2^k)*k!.
%F Stirling transform of A000165. - _Karol A. Penson_, Jan 25 2002
%F "AIJ" (ordered, indistinct, labeled) transform of 2, 2, 2, 2, ...
%F Recurrence: a(n) = 2*Sum_{k=1..n} binomial(n, k)*a(n-k), a(0)=1. - _Vladeta Jovovic_, Mar 27 2003
%F a(n) ~ (n-1)!/(3*(log(3/2))^n). - _Vaclav Kotesovec_, Aug 07 2013
%F a(n) = log(3/2)*Integral_{x>=0} floor(x)^n * (3/2)^(-x) dx. - _Peter Bala_, Feb 14 2015
%F E.g.f.: (x - log(3 - 2*exp(x)))/3. - _Ilya Gutkovskiy_, May 31 2018
%F 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
%t a[n_] := (1/3)*PolyLog[-n + 1, 2/3]; a[1]=1; Table[a[n], {n, 1, 18}] (* _Jean-François Alcover_, Jun 11 2012 *)
%t CoefficientList[Series[1/(3-2*Exp[x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Aug 07 2013 *)
%o (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 */
%o (PARI) my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(3 - 2*exp(x)))) \\ _Joerg Arndt_, Jan 15 2024
%o (Sage)
%o A004123 = lambda n: sum(stirling_number2(n-1,k)*(2^k)*factorial(k) for k in (0..n-1))
%o [A004123(n) for n in (1..18)] # _Peter Luschny_, Jan 18 2016
%Y Cf. A004121, A004122, A000165, A000670, A032033.
%Y Second row of array A094416 (generalized ordered Bell numbers).
%Y Equals 2 * A050351(n) for n>0.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_