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A004123
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Number of generalized weak orders on n points.
(Formerly M1975)
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15
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1, 2, 10, 74, 730, 9002, 133210, 2299754, 45375130, 1007179562, 24840104410, 673895590634, 19944372341530, 639455369290922, 22079273878443610, 816812844197444714, 32232133532123179930, 1351401783010933015082
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003
a(n) = 2^n A(n,3/2); A(n,x) the Eulerian polynomials. [From Peter Luschny, Aug 03 2010]
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagner, Carl G.; Enumeration of generalized weak orders. Arch. Math. (Basel) 39 (1982), no. 2, 147-152.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution.
C. G. Bower, Transforms
Foata, D. and Krattenthaler, C., Graphical Major Indices, II, Seminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
D. Foata and D. Zeilberger, [math/9406220] The Graphical Major Index
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FORMULA
| E.g.f. : 1/(3-2*exp(x)).
O.g.f.: Sum_{n>=0} 2^n*n!*x^(n+1)/Product_{k=0..n} (1-k*x). [From Paul D. Hanna, Jul 20 2011]
a(n) = sum(k^n*(2/3)^k, k = 0..infinity)/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 (vladeta(AT)eunet.rs), Mar 27 2003
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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 */
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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: A185971 A000698 A092881 * A086352 A005365 A191812
Adjacent sequences: A004120 A004121 A004122 * A004124 A004125 A004126
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net)
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