A258943 Exponential reversion of Fibonacci numbers A000045.

1, -1, 1, 2, -25, 132, -209, -5104, 77121, -550000, -1212751, 104886144, -1708324409, 11026947008, 204999609375, -8086017144832, 130002259664321, -256459066769664, -50556576336151391, 1626772400000000000, -22785408268305809049, -251717732965479463936

Offset

1,4

Links

Table of n, a(n) for n=1..22.

Mathoverflow, A special type of generating function for Fibonacci, 2020.

Formula

a(n) = 5^((n-1)/2)*Gamma(n/2 - x)/Gamma(1 - n/2 - x) where x = sqrt(5)*n/10. - Wolfgang Berndt, Feb 19 2020

Mathematica

length = 22; Range[length]! InverseSeries[Sum[Fibonacci[n] x^n/n!, {n, 1, length}] + O[x]^(length+1)][[3]]
Table[FullSimplify[5^((n-1)/2) * Gamma[(1 - 1/Sqrt[5])*n/2] / Gamma[1 - (1 + 1/Sqrt[5])*n/2]], {n, 1, 20}] (* Vaclav Kotesovec, Mar 23 2020 *)

Prog

(PARI) a(n) = {my(x=sqrt(5)*n/10); round(5^((n-1)/2)*gamma(n/2-x)/gamma(1-n/2-x))} \\ Wolfgang Berndt, Feb 19 2020
(PARI) seq(n)={Vec(serlaplace(serreverse(sum(k=1, n, fibonacci(k)*x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Feb 26 2020

Crossrefs

Cf. A000045.

Sequence in context: A226418 A196011 A080515 * A009381 A063264 A024533

Adjacent sequences: A258940 A258941 A258942 * A258944 A258945 A258946

Keyword

sign

Author

Vladimir Reshetnikov, Nov 07 2015

Status

approved