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

1, 2, 10, 78, 816, 10680, 167760, 3074400, 64391040, 1517201280, 39720844800, 1143895737600, 35937095040000, 1223098971494400, 44829605505484800, 1760481463732992000, 73744004937867264000, 3282093293695856640000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Laguerre transform of n!Fibonacci(n+1), A005442. - Paul Barry, Aug 08 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 510

Crossrefs

Apart from signs, row sums of A079461.

Cf. A005442, A122367.

Programs

Magma

m:=25; R:=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

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)/(1-3x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 07 2012 *)
Table[Fibonacci[2n+1] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

PARI

x='x+O('x^30); Vec(serlaplace((1-x)/(1-3*x+x^2))) \\ G. C. Greubel, May 23 2018

Formula

Recurrence: {a(0)=1, a(1)=2, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0.}

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

a(n) = Sum_{k=0..n} binomial(n,k)(n!/k!)*k!Fibonacci(k+1). - Paul Barry, Aug 08 2008

a(n) = n!*A122367(n). - R. J. Mathar, Nov 27 2011

Extensions

Edited by N. J. A. Sloane, May 29 2009