A263823 a(n) = n!*Sum_{k=0..n} Fibonacci(k-1)/k!, where Fibonacci(-1) = 1, Fibonacci(n) = A000045(n) for n>=0.

1, 1, 3, 10, 42, 213, 1283, 8989, 71925, 647346, 6473494, 71208489, 854501957, 11108525585, 155519358423, 2332790376722, 37324646028162, 634518982479741, 11421341684636935, 217005492008104349, 4340109840162091161, 91142306643403921146, 2005130746154886276158

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Eric Weisstein's MathWorld, Incomplete Gamma Function.

Eric Weisstein's MathWorld, Fibonacci Number.

Eric Weisstein's MathWorld, Golden Ratio.

Formula

a(n) = (Gamma(n+1, 1-phi)*exp(1-phi)*phi+Gamma(n+1, phi)*exp(phi)/phi)/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function, phi=(1+sqrt(5))/2.

a(n) = (phi^(n-1)*hypergeom([1,-n], [], 1-phi]-(-phi)^(1-n)*hypergeom([1,-n], [], phi))/sqrt(5).

Gamma(n+1, phi)*exp(phi) = A111139(n)*phi + a(n).

E.g.f.: (exp(phi*x)/phi+exp(-x/phi)*phi)/(sqrt(5)*(1-x)) = exp(x/2)*(cosh(x*sqrt(5)/2)-sinh(x*sqrt(5)/2)/sqrt(5))/(1-x).

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

a(n) ~ 2*exp(phi-n)*n^(n+1/2)*(1+exp(-sqrt(5))*phi^2)*sqrt(Pi/10)/phi.

0 = a(n)*(+a(n+1) + a(n+2) - 4*a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) + 3*a(n+2) - 5*a(n+3) + a(n+4)) + a(n+2)*(+2*a(n+2) - a(n+4)) + a(n+3)*(+a(n+3)) if n>=0. - Michael Somos, Oct 30 2015

Example

For n = 3, a(3) = 3!*(Fibonacci(-1)/0! + Fibonacci(0)/1! + Fibonacci(1)/2! + Fibonacci(2)/3!) = 6*(1 + 0 + 1/2 + 1/6) = 10.
For n = 5, Gamma(5+1, phi)*exp(phi) = 120*sqrt(5) + 333 = 240*phi + 213, so a(5) = 213.
G.f. = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 213*x^5 + 1283*x^6 + 8989*x^7 + 71925*x^8 + ...

Mathematica

Table[n! Sum[Fibonacci[k-1]/k!, {k, 0, n}], {n, 0, 22}]
Round@Table[(E^(1-GoldenRatio) GoldenRatio Gamma[n+1, 1-GoldenRatio] + E^GoldenRatio Gamma[n+1, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 22}]

Crossrefs

Cf. A000045, A111139.

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

Sequence in context: A030903 A030816 A030964 * A030867 A186360 A352856

Adjacent sequences: A263820 A263821 A263822 * A263824 A263825 A263826

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 27 2015

Status

approved