A111139 a(n) = n!*Sum_{k=0..n} Fibonacci(k)/k!.

0, 1, 3, 11, 47, 240, 1448, 10149, 81213, 730951, 7309565, 80405304, 964863792, 12543229529, 175605213783, 2634078207355, 42145251318667, 716469272418936, 12896446903543432, 245032491167329389, 4900649823346594545

Offset: 0

Author

Vladeta Jovovic, Oct 17 2005

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A000045, A263823.

Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

Programs

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 *)

PARI

vector(100, n, n--; n!*sum(k=0, n, fibonacci(k)/k!)) \\ Altug Alkan, Oct 28 2015

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)