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A004123 Number of generalized weak orders on n points.
(Formerly M1975)
24
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

a(n) = 2^n A(n,3/2); A(n,x) the Eulerian polynomials. [Peter Luschny, Aug 03 2010]

REFERENCES

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

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.

Carl G. Wagner, Enumeration of generalized weak orders, Arch. Math. (Basel) 39 (1982), no. 2, 147-152.

FORMULA

E.g.f.(for shifted sequence with offset 0): 1/(3-2*exp(x)).

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^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, 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)*int {x = 0..inf} floor(x)^n * (3/2)^(-x) dx. - Peter Bala, Feb 14 2015

MATHEMATICA

a[n_] := (1/3)*PolyLog[-n + 1, 2/3]; a[1]=1; Table[a[n], {n, 1, 18}] (* Jean-Fran├žois 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: A185971 A000698 A092881 * A086352 A005365 A191812

Adjacent sequences:  A004120 A004121 A004122 * A004124 A004125 A004126

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Christian G. Bower

STATUS

approved

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Last modified March 28 12:08 EDT 2017. Contains 284186 sequences.