OFFSET
0,3
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
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
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 *)
PROG
(Sage) [stirling_number1(i, 2)*i for i in range(0, 32)] # Zerinvary Lajos, Jun 27 2008
(PARI) x='x+O('x^66); concat([0, 0], Vec(serlaplace(-log(-1/(-1+x))*x/(-1+x)))) \\ Joerg Arndt, May 06 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
New name using e.g.f., Vaclav Kotesovec, Feb 25 2014
STATUS
approved