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%I A260787 #22 Sep 05 2015 09:54:44
%S A260787 1,2,6,15,38,89,210,474,1065,2339,5091,10919,23230,48887,102126,
%T A260787 211599,435561,890617,1810786,3661118,7365473,14747049,29397160,
%U A260787 58356179,115392801,227332038,446304671,873298579,1703463864,3312873935,6424553973,12425158365,23968214357,46120280910,88535346223
%N A260787 G.f.: Product_{k>=1} 1/(1-x^k)^Fibonacci(k+2).
%C A260787 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 A260787 Vaclav Kotesovec, <a href="/A260787/b260787.txt">Table of n, a(n) for n = 0..4450</a>
%H A260787 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. See F_H.
%H A260787 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 A260787 Vaclav Kotesovec, <a href="/A034691/a034691_1.pdf">Asymptotics of sequence A034691</a>
%F A260787 a(n) ~ phi^(n+1/2) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 7/10 + 2*5^(-1/4)*phi*sqrt(n) + s), where s = Sum_{k>=2} (1 + 2*phi^k) / ((phi^(2*k) - phi^k - 1)*k) = 1.39069800276768443926918973402733105305129194986259856042723... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 06 2015
%t A260787 CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k+2], {k, 1, 20}], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Aug 05 2015 *)
%Y A260787 Cf. A000045, A034691, A166861, A200544.
%K A260787 nonn
%O A260787 0,2
%A A260787 _N. J. A. Sloane_, Aug 05 2015

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