A308940 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} Fibonacci(k)*x^k/k!).

1, 1, 3, 14, 85, 645, 5878, 62495, 759351, 10379878, 157652085, 2633903669, 48005235886, 947849607015, 20154635314591, 459170181891230, 11158379672316837, 288109467764819749, 7876576756719778854, 227299554620022188879, 6904560742996004248135

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

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

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

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

Mathematica

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

Crossrefs

Cf. A000045, A097597, A215928.

Sequence in context: A331608 A331615 A317060 * A276313 A074520 A127715

Adjacent sequences: A308937 A308938 A308939 * A308941 A308942 A308943

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 01 2019

Status

approved