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

0,3

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.

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A234953 A344051 A025168 * A050351 A129137 A357397

Adjacent sequences: A084355 A084356 A084357 * A084359 A084360 A084361

Keyword

nonn

Author

N. J. A. Sloane, Jun 22 2003

Status

approved