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A007553
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Logarithmic transform of Fibonacci numbers.
(Formerly M4329)
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6
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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
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A329008 A005692 A080798 * A294474 A248277 A002019
Adjacent sequences: A007550 A007551 A007552 * A007554 A007555 A007556
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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