A007553 Logarithmic transform of Fibonacci numbers.
1, 1, 1, 1, 7, 5, 85, 335, 1135, 15245, 13475, 717575, 4256825, 29782325, 525045275, 243258625, 56809006625, 415670267875, 5068080417875, 104229929847625, 60861649495625, 20784245979986875, 169274937975443125, 3318579283890780625, 75028912866554839375
Offset: 1
Keywords
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..340
- 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
Programs
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Maple
b:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*t(n-j)* binomial(n, j)*b(j), j=1..n-1)/n))(i->(<<0|1>, <1|1>>^i)[2, 2]) end: a:= n-> abs(b(n)): seq(a(n), n=1..30); # Alois P. Heinz, Mar 06 2018 -
Mathematica
FullSimplify[Abs[Rest[CoefficientList[Series[-2*x/(1+Sqrt[5]) - Log[5+Sqrt[5]] + Log[2+(3+Sqrt[5])*E^(Sqrt[5]*x)], {x, 0, 15}], x] * Range[0, 15]!]]] (* Vaclav Kotesovec, Jun 24 2014 *) -
Maxima
b(n):=if n=0 then 1 else b(n-1)-sum(b(i)*b(n-1-i)*binomial(n-1,i),i,1,n-2); a(n):=if n=0 then 0 else abs(b(n-1)); # Vladimir Kruchinin, Nov 03 2011
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Maxima
b(n):=if n=1 then 1 else sum((n+k-1)!*sum(((-1)^(j)/(k-j)!*sum(((sqrt(5)+1)^(n+j-i-1)*5^((i-j)/2)*stirling1(i,j)*2^(-n-j+i+1)*binomial(n+j-2,i-1))/i!,i,j,n+j-1)),j,1,k),k,1,n-1); a(n):=if n=1 then 1 else abs(b(n-1)); makelist(ratsimp(a(n)),n,1,10); # Vladimir Kruchinin, Nov 17 2012
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Sage
@CachedFunction def c(n,k) : if n==k: return 1 if k<1 or k>n: return 0 return ((n-k)//2+1)*c(n-1,k-1)-2*k*c(n-1,k+1) @CachedFunction def A007553(n): return abs(add(c(n,k) for k in (0..n))) [A007553(n) for n in (0..25)] # Peter Luschny, Jun 10 2014
Formula
b(n) = b(n-1) - Sum_{i=1..n-2} b(i)*b(n-1-i)*binomial(n-1,i), b(0)=1. a(n+1) = abs(b(n)). - Vladimir Kruchinin, Nov 03 2011
Let e.g.f. E(x) = log(1 + Sum_{n>=1} Fibonacci(n+1)*x^n/n!), then g.f. A(x)=x*(1+1/Q(0)), where Q(k) = 1/(x*(k+1)) + 1 + 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013
Let F(x) = log(Sum_{n>=0} Fibonacci(n+1)*x^n/n!) be the e.g.f., produce the sequence [1,1,-1,-1,7,-5,-85,...], then g.f. A(x)= 1 + x/Q(0), where Q(k) = 1 + x*(2*k+1) + x^2*(2*k+1)*(2*k+2)/(1 + x*(2*k+2) + x^2*(2*k+2)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013
a(n) ~ 2*(n-1)! * abs(cos(n*arctan(Pi/log(2/(3+sqrt(5)))))) * (5/(Pi^2+log(2/(3+sqrt(5)))^2))^(n/2). - Vaclav Kotesovec, Jun 24 2014
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