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A007553 Logarithmic transform of Fibonacci numbers.
(Formerly M4329)
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
The coefficients of the e.g.f. log(Sum_{n>=0} Fibonacci(n+1)*x^n/n!) produce the sequence [1,1,-1,-1,7,-5,-85,...], offset 0. - Peter Bala, Jan 19 2011
The series reversion of Sum_{n>=1} Fibonacci(n)*x^n/n is an e.g.f. whose coefficient sequence [1,-1,-1,7,-5,-85,335,1135,...] (offset 1) appears to be a signed version of this sequence. - Peter Bala, Jan 19 2011
E.g.f. A(x), A(x)=x*B(x) satisfies the differential equation B'(x) = 1 + B(x) - B(x)*B(x). - Vladimir Kruchinin, Nov 03 2011
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
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
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 *)
PROG
(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
(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
(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
CROSSREFS
Sequence in context: A329008 A005692 A080798 * A294474 A248277 A002019
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)