A052881 Expansion of e.g.f. log(1/(1-x))*x/(1-x).

0, 0, 2, 9, 44, 250, 1644, 12348, 104544, 986256, 10265760, 116915040, 1446526080, 19323757440, 277238626560, 4251984710400, 69426608025600, 1202482800691200, 22021300630425600, 425162773111910400

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, arXiv preprint arXiv:1412.0345 [math.CO], 2014 and J. Int. Seq. 18 (2015) 15.10.1.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 853

Programs

Maple

spec := [S,{B=Sequence(Z,1 <= card),C=Cycle(Z),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a:=n->abs(Stirling1(n,2))*n: seq(a(n), n=0..19); # Zerinvary Lajos, Oct 05 2007
A052881 := n -> `if`(n<2,0,2*(n-1)*(n-1)!*hypergeom([1,1,2-n],[2,n+1],-1)):
seq(simplify(A052881(n)),n=0..19); # Peter Luschny, Jun 11 2016

Mathematica

Table[n!*SeriesCoefficient[-Log[-1/(-1+x)]*x/(-1+x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)
With[{nn=20},CoefficientList[Series[Log[1/(1-x)] x/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 19 2022 *)

PARI

x='x+O('x^66); concat([0,0],Vec(serlaplace(-log(-1/(-1+x))*x/(-1+x)))) \\ Joerg Arndt, May 06 2013

Sage

[stirling_number1(i,2)*i for i in range(0,32)] # Zerinvary Lajos, Jun 27 2008

Formula

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

Recurrence: a(1)=0, a(2)=2, (n^3+3*n^2+2*n)*a(n)+(-5*n-2*n^2-2)*a(n+1)+(n+1)*a(n+2) =0.

a(n) = n!*Sum 1/i, i = 1..(n-1) = s(n, 2)-(n-1)! = n*s(n-1, 2) = n*a(n-1) + (n-1)! + (n-2)! = A000142(n)*A001008(n-1)/A002805(n-1) = A000254(n)-A000142(n-1) = A000027(n)*A000254(n-1) = a(n-1)*A000027(n) + A001048(n-1). - Henry Bottomley, May 05 2001

a(n) ~ n!*log(n)*(1+gamma/log(n)), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 09 2012

a(n) = 2*(n-1)*(n-1)!*hypergeom([1,1,2-n], [2,n+1], -1) for n>=2. - Peter Luschny, Jun 11 2016

Extensions

New name using e.g.f., Vaclav Kotesovec, Feb 25 2014