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A166861
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Euler transform of Fibonacci numbers.
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33
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1, 1, 2, 4, 8, 15, 30, 56, 108, 203, 384, 716, 1342, 2487, 4614, 8510, 15675, 28749, 52652, 96102, 175110, 318240, 577328, 1045068, 1888581, 3406455, 6134530, 11029036, 19799363, 35490823, 63531134, 113570988, 202767037, 361565865, 643970774, 1145636750
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OFFSET
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0,3
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COMMENTS
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In general, the sequence with g.f. Product_{k>=1} 1/(1-x^k)^Fibonacci(k+z), where z is nonnegative integer, is asymptotic to phi^(n + z/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp((phi/10 - 1/2) * Fibonacci(z) - Fibonacci(z+1)/10 + 2 * 5^(-1/4) * phi^(z/2) * sqrt(n) + s), where s = Sum_{k>=2} (Fibonacci(z) + Fibonacci(z+1) * phi^k) / ((phi^(2*k) - phi^k - 1)*k) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..4550
Loic Foissy, The Hopf algebra of Fliess operators and its dual pre-Lie algebra, 2013.
W. S. Gray, K. Ebrahimi-Fard, Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra, arXiv:1411.0222 [math.OC], 2014.
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
Vaclav Kotesovec, Asymptotics of sequence A034691
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FORMULA
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G.f.: Product_{k>0} 1/(1 - x^k)^Fibonacci(k).
a(n) ~ phi^n / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(-1/10 + 2*5^(-1/4)*sqrt(n) + s), where s = Sum_{k>=2} phi^k / ((phi^(2*k) - phi^k - 1)*k) = 0.600476601392575912969719494850393576083765123939643511355547131467... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015
G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 29 2018
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 30*x^6 + 56*x^7 + 108*x^8 + 203*x^9 + ...
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MAPLE
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F:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
F(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jan 12 2017
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MATHEMATICA
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CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 05 2015 *)
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PROG
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(PARI) ET(v)=Vec(prod(k=1, #v, 1/(1-x^k+x*O(x^#v))^v[k]))
ET(vector(40, n, fibonacci(n)))
(SageMath)
def EulerTransform(a):
@cached_function
def b(n):
if n == 0: return 1
s = sum(sum(d * a(d) for d in divisors(j)) * b(n-j) for j in (1..n))
return s//n
return b
a = BinaryRecurrenceSequence(1, 1)
b = EulerTransform(a)
print([b(n) for n in range(36)]) # Peter Luschny, Nov 11 2020
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CROSSREFS
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Cf. A000045, A034691, A109509, A200544, A260787, A261031, A261050, A260916, row sums of A337009.
Sequence in context: A301480 A217777 A034338 * A026023 A077596 A091865
Adjacent sequences: A166858 A166859 A166860 * A166862 A166863 A166864
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KEYWORD
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nonn
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AUTHOR
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Franklin T. Adams-Watters, Oct 21 2009
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EXTENSIONS
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First formula corrected by Vaclav Kotesovec, Aug 05 2015
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STATUS
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approved
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