OFFSET
0,2
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
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
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 *)
PROG
(PARI) x='x+O('x^30); Vec(serlaplace((1-x)/(1-3*x+x^2))) \\ G. C. Greubel, May 23 2018
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Edited by N. J. A. Sloane, May 29 2009
STATUS
approved