A112005 - OEIS

%I #23 Jun 08 2021 08:22:22

%S 0,1,0,1,-2,4,-17,82,-384,2189,-14850,107404,-845537,7400482,

%T -70093256,709888645,-7721333538,89774204756,-1107347563761,

%U 14456268008050,-199350032354000,2893615098314941,-44089764970860290,703841452185590236,-11747695951762870497

%N Logarithmic transform of Fibonacci numbers A000045.

%H Alois P. Heinz, <a href="/A112005/b112005.txt">Table of n, a(n) for n = 0..200</a>

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>

%F 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.

%F 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

%p a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*t(n-j)*

%p       binomial(n, j)*a(j), j=1..n-1)/n))(i->(<<0|1>, <1|1>>^i)[1, 2])

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 06 2018

%t 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 *)

%Y Cf. A007553, A112006, A256180.

%K sign,easy

%O 0,5

%A _Wolfdieter Lang_, Sep 12 2005