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%I #46 Nov 11 2020 05:25:06
%S 1,1,2,4,8,15,30,56,108,203,384,716,1342,2487,4614,8510,15675,28749,
%T 52652,96102,175110,318240,577328,1045068,1888581,3406455,6134530,
%U 11029036,19799363,35490823,63531134,113570988,202767037,361565865,643970774,1145636750
%N Euler transform of Fibonacci numbers.
%C 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
%H Vaclav Kotesovec, <a href="/A166861/b166861.txt">Table of n, a(n) for n = 0..4550</a>
%H Loic Foissy, <a href="http://hal.archives-ouvertes.fr/docs/00/80/85/13/PDF/control.pdf">The Hopf algebra of Fliess operators and its dual pre-Lie algebra</a>, 2013.
%H W. S. Gray, K. Ebrahimi-Fard, <a href="http://arxiv.org/abs/1411.0222">Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra</a>, arXiv:1411.0222 [math.OC], 2014.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1508.01796">Asymptotics of the Euler transform of Fibonacci numbers</a>, arXiv:1508.01796 [math.CO], Aug 07 2015
%H Vaclav Kotesovec, <a href="/A034691/a034691_1.pdf">Asymptotics of sequence A034691</a>
%F G.f.: Product_{k>0} 1/(1 - x^k)^Fibonacci(k).
%F 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
%F G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k - x^(2*k)))). - _Ilya Gutkovskiy_, May 29 2018
%e 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 + ...
%p F:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p F(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jan 12 2017
%t CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k], {k, 1, 40}], {x, 0, 40}], x] (* _Vaclav Kotesovec_, Aug 05 2015 *)
%o (PARI) ET(v)=Vec(prod(k=1,#v,1/(1-x^k+x*O(x^#v))^v[k]))
%o ET(vector(40,n,fibonacci(n)))
%o (SageMath)
%o def EulerTransform(a):
%o @cached_function
%o def b(n):
%o if n == 0: return 1
%o s = sum(sum(d * a(d) for d in divisors(j)) * b(n-j) for j in (1..n))
%o return s//n
%o return b
%o a = BinaryRecurrenceSequence(1, 1)
%o b = EulerTransform(a)
%o print([b(n) for n in range(36)]) # _Peter Luschny_, Nov 11 2020
%Y Cf. A000045, A034691, A109509, A200544, A260787, A261031, A261050, A260916, row sums of A337009.
%K nonn
%O 0,3
%A _Franklin T. Adams-Watters_, Oct 21 2009
%E First formula corrected by _Vaclav Kotesovec_, Aug 05 2015
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