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A052554 Expansion of e.g.f.: (1-x)/(1 - x - x^2). 3

%I

%S 1,0,2,6,48,360,3600,40320,524160,7620480,123379200,2195424000,

%T 42631142400,896690995200,20312541849600,492993236736000,

%U 12762901831680000,351063491530752000,10224590808047616000

%N Expansion of e.g.f.: (1-x)/(1 - x - x^2).

%C Number of ways to use the elements of {1,..,n} once each to form a sequence of lists, each having length at least 2. - Bob Proctor, Apr 19 2005

%H G. C. Greubel, <a href="/A052554/b052554.txt">Table of n, a(n) for n = 0..415</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=493">Encyclopedia of Combinatorial Structures 493</a>

%H Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F a(n) = n*a(n-1) + n*(n-1)*a(n-2), with a(0)=1, a(1)=0.

%F a(n) = Sum(1/5*(-1+3*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+_Z^2))*n!.

%F a(n) = n!*Fibonacci(n-1) for n >= 1. - Bob Proctor, Apr 19 2005

%F a(0) = 1; a(n) = Sum_{k=2..n} binomial(n,k) * k! * a(n-k). - _Ilya Gutkovskiy_, Feb 09 2020

%F a(n)=n!*A212804(n). - _R. J. Mathar_, Jun 03 2022

%p spec := [S,{S=Sequence(Prod(Z,Z,Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t With[{m=20}, CoefficientList[Series[(1-x)/(1-x-x^2), {x,0,m}], x]* Range[0, m]!] (* _G. C. Greubel_, May 07 2019 *)

%o (PARI) my(x='x+O('x^20)); Vec(serlaplace( (1-x)/(1-x-x^2) )) \\ _G. C. Greubel_, May 07 2019

%o (MAGMA) [1] cat [Fibonacci(n-1)*Factorial(n): n in [1..20]] // _G. C. Greubel_, May 07 2019

%o (Sage) [1]+[fibonacci(n-1)*factorial(n) for n in (1..20)] # _G. C. Greubel_, May 07 2019

%o (GAP) a:=[0,2];; for n in [3..20] do a[n]:=n*a[n-1]+n*(n-1)*a[n-2]; od; Concatenation([1], a); # _G. C. Greubel_, May 07 2019

%K easy,nonn

%O 0,3

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

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Last modified August 18 04:45 EDT 2022. Contains 356204 sequences. (Running on oeis4.)