login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A308940 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} Fibonacci(k)*x^k/k!). 0

%I

%S 1,1,3,14,85,645,5878,62495,759351,10379878,157652085,2633903669,

%T 48005235886,947849607015,20154635314591,459170181891230,

%U 11158379672316837,288109467764819749,7876576756719778854,227299554620022188879,6904560742996004248135

%N Expansion of e.g.f. 1 / (1 - Sum_{k>=1} Fibonacci(k)*x^k/k!).

%F E.g.f.: sqrt(5)/(sqrt(5) - 2*exp(x/2)*sinh(sqrt(5)*x/2)).

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Fibonacci(k) * a(n-k).

%F a(n) ~ n! * 5^((n+1)/2) * (exp(2*r) - 1) / ((sqrt(5) - 1 + (1 + sqrt(5))*exp(2*r)) * 2^n * r^(n+1)), where r = 0.7361181605960590527095268838693519750655284224... is the root of the equation exp(2*r) = 1 + sqrt(5)*exp(r*(1 - 1/sqrt(5))). - _Vaclav Kotesovec_, Jul 01 2019

%t nmax = 20; CoefficientList[Series[Sqrt[5]/(Sqrt[5] - 2 Exp[x/2] Sinh[Sqrt[5] x/2]), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Fibonacci[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

%Y Cf. A000045, A097597, A215928.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 01 2019

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 15 02:56 EDT 2021. Contains 345042 sequences. (Running on oeis4.)