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A052568 E.g.f.: (1-x)/(1-3*x+x^2).

%I A052568 #32 Sep 08 2022 08:44:59
%S A052568 1,2,10,78,816,10680,167760,3074400,64391040,1517201280,39720844800,
%T A052568 1143895737600,35937095040000,1223098971494400,44829605505484800,
%U A052568 1760481463732992000,73744004937867264000,3282093293695856640000
%N A052568 E.g.f.: (1-x)/(1-3*x+x^2).
%C A052568 Laguerre transform of n!Fibonacci(n+1), A005442. - _Paul Barry_, Aug 08 2008
%H A052568 G. C. Greubel, <a href="/A052568/b052568.txt">Table of n, a(n) for n = 0..385</a>
%H A052568 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=510">Encyclopedia of Combinatorial Structures 510</a>
%F A052568 Recurrence: {a(0)=1, a(1)=2, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0.}
%F A052568 a(n) = Sum(1/5*(1+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^2-3*_Z+1))*n!.
%F A052568 a(n) = Sum_{k=0..n} binomial(n,k)(n!/k!)*k!Fibonacci(k+1). - _Paul Barry_, Aug 08 2008
%F A052568 a(n) = n!*A122367(n). - _R. J. Mathar_, Nov 27 2011
%p A052568 spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052568 a:= n-> n! * (Matrix([[1,1]]). Matrix([[3,1], [ -1,0]])^n)[1,1]: seq(a(n), n=0..20); # _Alois P. Heinz_, Jun 01 2009
%t A052568 With[{nn=20},CoefficientList[Series[(1-x)/(1-3x+x^2),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 07 2012 *)
%t A052568 Table[Fibonacci[2n+1] n!, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 29 2015 *)
%o A052568 (PARI) x='x+O('x^30); Vec(serlaplace((1-x)/(1-3*x+x^2))) \\ _G. C. Greubel_, May 23 2018
%o A052568 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)/(1-3*x+x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 23 2018
%Y A052568 Apart from signs, row sums of A079461.
%Y A052568 Cf. A005442, A122367.
%K A052568 easy,nonn
%O A052568 0,2
%A A052568 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052568 Edited by _N. J. A. Sloane_, May 29 2009