OFFSET
0,3
COMMENTS
Eigensequence of a triangle with the Fibonacci series as the left border, the natural numbers (1, 2, 3, ...) as the right border; and the rest zeros. - Gary W. Adamson, Aug 01 2016
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's MathWorld, Incomplete Gamma Function.
Eric Weisstein's MathWorld, Fibonacci Number.
Eric Weisstein's MathWorld, Golden Ratio.
FORMULA
E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2)/(1-x).
Recurrence: a(n) = (n+1)*a(n-1) - (n-2)*a(n-2) - (n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2*sqrt(e/5)*sinh(sqrt(5)/2)*n!. - Vaclav Kotesovec, Oct 18 2012
From Vladimir Reshetnikov, Oct 27 2015: (Start)
Let phi=(1+sqrt(5))/2.
a(n) = (phi^n*hypergeom([1,-n], [], 1-phi)-(1-phi)^n*hypergeom([1,-n], [], phi))/sqrt(5).
a(n) = (exp(phi)*Gamma(n+1, phi)-exp^(1-phi)*Gamma(n+1, 1-phi))/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function.
Gamma(n+1, phi)*exp(phi) = a(n)*phi + A263823(n).
a(n) ~ exp(phi-n)*n^(n+1/2)*sqrt(2*Pi/5)*(1-exp(-sqrt(5))).
(End)
MAPLE
a:=n->sum(fibonacci (j)*n!/j!, j=0..n):seq(a(n), n=0..20); # Zerinvary Lajos, Mar 19 2007
MATHEMATICA
f[n_] := n!*Sum[Fibonacci[k]/k!, {k, 0, n}]; Table[ f[n], {n, 0, 20}] (* or *)
Simplify[ Range[0, 20]!CoefficientList[ Series[2/Sqrt[5]*Exp[x/2]*Sinh[Sqrt[5]*x/2]/(1 - x), {x, 0, 20}], x]] (* Robert G. Wilson v, Oct 21 2005 *)
Module[{nn=20, fibs, fct}, fct=Range[0, nn]!; fibs=Accumulate[ Fibonacci[ Range[ 0, nn]]/fct]; Times@@@Thread[{fct, fibs}]] (* Harvey P. Dale, Feb 19 2014 *)
Round@Table[(E^GoldenRatio Gamma[n+1, GoldenRatio] - E^(1-GoldenRatio) Gamma[n+1, 1-GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 27 2015 *)
PROG
(PARI) vector(100, n, n--; n!*sum(k=0, n, fibonacci(k)/k!)) \\ Altug Alkan, Oct 28 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Oct 17 2005
STATUS
approved