A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

1, 2, 9, 62, 572, 6604, 91526, 1480044, 27353448, 568731648, 13138994112, 333895239072, 9256507508112, 278000959058016, 8991458660924112, 311585506208924064, 11517363473843526912, 452332548042633835776

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

a(n) is the number of ways to seat n people at circular tables, then linearly order the tables, then designate some (possibly all or none) of the tables at which only one person is seated. a(2) = 9 because we have: (1)(2), (1')(2), (1)(2'), (1')(2'), (2)(1), (2')(1), (2)(1'), (2')(1'), (1,2). Cf. A007840. - Geoffrey Critzer, Nov 05 2013

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..395
• W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 785
• Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Cascaded Analytic Nonlinear Systems, 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, December 12-15, 2011, pp. 3830-3835.
• M. Thitsa and W. S. Gray, On the radius of convergence of cascaded analytic nonlinear systems: The SISO case, System Theory (SSST), 2011 IEEE 43rd Southeastern Symposium on, 14-16 March 2011, pp. 30-36.
• Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, 2012, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012

Crossrefs

Cf. A030178.

Cf. A367851, A367852, A367853.

Cf. A367845, A367846, A367847.

Programs

Maple

spec := [S,{C=Cycle(Z),B=Union(C,Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

CoefficientList[Series[1/(1-x+Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

Formula

E.g.f.: -1/(-1+x+log(-1/(-1+x))).

a(n) ~ n! * (1/(1-LambertW(1)))^n/(1/LambertW(1)-LambertW(1)). - Vaclav Kotesovec, Oct 01 2013

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

Extensions

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013