A084358 Lists of sets of lists.

1, 1, 5, 37, 363, 4441, 65133, 1114009, 21771851, 478658101, 11692343253, 314170940293, 9209104364331, 292435635165649, 10000637145321917, 366427621403088433, 14321135069200849515, 594696814358067968461, 26147933188037724372069

Offset: 0

Author

N. J. A. Sloane, Jun 22 2003

Keywords

• nonn

Comments

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

References

• 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7
• M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Programs

Magma

m:=25; R:=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

Maple

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)];

Mathematica

With[{nn=20},CoefficientList[Series[1/(2-Exp[x/(1-x)]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 16 2013 *)

PARI

x='x+O('x^30); Vec(serlaplace(1/(2-exp(x/(1-x))))) \\ G. C. Greubel, May 16 2018

Formula

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