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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..inf} 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..inf} 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

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

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..inf} 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

Adjacent sequences:  A007550 A007551 A007552 * A007554 A007555 A007556

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified May 18 01:26 EDT 2021. Contains 343992 sequences. (Running on oeis4.)