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

%I A052881 #49 Aug 13 2024 01:52:13
%S A052881 0,0,2,9,44,250,1644,12348,104544,986256,10265760,116915040,
%T A052881 1446526080,19323757440,277238626560,4251984710400,69426608025600,
%U A052881 1202482800691200,22021300630425600,425162773111910400
%N A052881 Expansion of e.g.f. log(1/(1-x))*x/(1-x).
%C A052881 Previous name was: A simple grammar.
%H A052881 Vincenzo Librandi, <a href="/A052881/b052881.txt">Table of n, a(n) for n = 0..200</a>
%H A052881 Matt Davis, <a href="http://arxiv.org/abs/1412.0345">Quadrant Marked Mesh Patterns and the r-Stirling Numbers</a>, arXiv preprint arXiv:1412.0345 [math.CO], 2014  and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Davis/davis3.html">J. Int. Seq. 18 (2015) 15.10.1</a>.
%H A052881 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=853">Encyclopedia of Combinatorial Structures 853</a>
%F A052881 E.g.f.: -log(-1/(-1+x))*x/(-1+x).
%F A052881 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.
%F A052881 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
%F A052881 a(n) ~ n!*log(n)*(1+gamma/log(n)), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 09 2012
%F A052881 a(n) = 2*(n-1)*(n-1)!*hypergeom([1,1,2-n], [2,n+1], -1) for n>=2. - _Peter Luschny_, Jun 11 2016
%p A052881 spec := [S,{B=Sequence(Z,1 <= card),C=Cycle(Z),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052881 a:=n->abs(Stirling1(n,2))*n: seq(a(n), n=0..19); # _Zerinvary Lajos_, Oct 05 2007
%p A052881 A052881 := n -> `if`(n<2,0,2*(n-1)*(n-1)!*hypergeom([1,1,2-n],[2,n+1],-1)):
%p A052881 seq(simplify(A052881(n)),n=0..19); # _Peter Luschny_, Jun 11 2016
%t A052881 Table[n!*SeriesCoefficient[-Log[-1/(-1+x)]*x/(-1+x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 09 2012 *)
%t A052881 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 *)
%o A052881 (Sage) [stirling_number1(i,2)*i for i in range(0,32)] # _Zerinvary Lajos_, Jun 27 2008
%o A052881 (PARI) x='x+O('x^66); concat([0,0],Vec(serlaplace(-log(-1/(-1+x))*x/(-1+x)))) \\ _Joerg Arndt_, May 06 2013
%K A052881 easy,nonn
%O A052881 0,3
%A A052881 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052881 New name using e.g.f., _Vaclav Kotesovec_, Feb 25 2014