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A084358
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Lists of sets of lists.
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9
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1, 1, 5, 37, 363, 4441, 65133, 1114009, 21771851, 478658101, 11692343253, 314170940293, 9209104364331, 292435635165649, 10000637145321917, 366427621403088433, 14321135069200849515, 594696814358067968461, 26147933188037724372069
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OFFSET
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0,3
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COMMENTS
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This sequence and -A000262 with the first term set to 1 form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Oct 21 2007
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REFERENCES
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T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
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LINKS
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FORMULA
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a(n) = n!*Lag{n,(.)!*Lag[.,P(.,2),0],-1} = P(n,2) - n*P(n-1,2) umbrally, where P(j,t) are the polynomials in A131758 and Lag(n,x,a) are the associated Laguerre polynomials of order a; that is, the sequence is given by an iterated combinatorial Laguerre transform, of mixed order, of a set of polynomials related to the polylogarithms, which reduces to a simple finite difference. - Tom Copeland, Sep 30 2007
E.g.f.: 1/(2-exp(x/(1-x))). Lah transform of preferential arrangements: Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*A000670(k). - Vladeta Jovovic, Sep 28 2003
a(n) ~ n! * (1+log(2))^(n-1) / (2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 08 2013
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MAPLE
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with(combstruct); SeqSetSeqL := [T, {T=Sequence(S), S=Set(U, card >= 1), U=Sequence(Z, card >=1)}, labeled]; [seq(count(%, size=j), j=1..12)];
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[1/(2-Exp[x/(1-x)]), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Apr 16 2013 *)
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PROG
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(PARI) x='x+O('x^30); Vec(serlaplace(1/(2-exp(x/(1-x))))) \\ G. C. Greubel, May 16 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(2-Exp(x/(1-x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 16 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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