A112005 Logarithmic transform of Fibonacci numbers A000045.

0, 1, 0, 1, -2, 4, -17, 82, -384, 2189, -14850, 107404, -845537, 7400482, -70093256, 709888645, -7721333538, 89774204756, -1107347563761, 14456268008050, -199350032354000, 2893615098314941, -44089764970860290, 703841452185590236, -11747695951762870497

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• sign
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms
• Index entries for sequences related to logarithmic numbers

Crossrefs

Cf. A007553, A112006, A256180.

Programs

Maple

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*t(n-j)*
      binomial(n, j)*a(j), j=1..n-1)/n))(i->(<<0|1>, <1|1>>^i)[1, 2])
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

Mathematica

FullSimplify[CoefficientList[Series[Log[1 + 2*E^(x/2)*Sinh[Sqrt[5]*x/2] / Sqrt[5]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Sep 04 2014 *)

Formula

E.g.f. log(1 + A(x)) with the e.g.f. A(x):=exp(x/2)*sinh(sqrt(5)*x/2)/(sqrt(5)/2) of A000045.

a(n) ~ -(n-1)! / r^n, where r = -1.37807491378452630283968362340785266756... is the root of the equation 2*(5-3*sqrt(5))*r + (sqrt(5)-5) * (log(5/4) + 2*log(1-coth(sqrt(5)*r/2))) = 0. - Vaclav Kotesovec, Sep 04 2014