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%I #40 Sep 08 2022 08:45:11
%S 1,1,5,37,363,4441,65133,1114009,21771851,478658101,11692343253,
%T 314170940293,9209104364331,292435635165649,10000637145321917,
%U 366427621403088433,14321135069200849515,594696814358067968461,26147933188037724372069
%N Lists of sets of lists.
%C 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
%D T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
%D T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%H Vincenzo Librandi, <a href="/A084358/b084358.txt">Table of n, a(n) for n = 0..200</a>
%H T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/He/he51.html">A symbolic operator approach to power series transformation-expansion formulas</a>, JIS 11 (2008) 08.2.7
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3
%H N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%F 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
%F 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
%F a(n) ~ n! * (1+log(2))^(n-1) / (2*(log(2))^(n+1)). - _Vaclav Kotesovec_, Oct 08 2013
%p 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)];
%t With[{nn=20},CoefficientList[Series[1/(2-Exp[x/(1-x)]),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Apr 16 2013 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(2-exp(x/(1-x))))) \\ _G. C. Greubel_, May 16 2018
%o (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
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 22 2003
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