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A261050 Expansion of Product_{k>=1} (1+x^k)^(Fibonacci(k)).

%I A261050 #13 May 29 2018 16:37:29
%S A261050 1,1,1,3,5,10,19,36,67,127,236,438,811,1496,2750,5046,9224,16827,
%T A261050 30630,55623,100803,182342,329205,593326,1067591,1917885,3440207,
%U A261050 6162004,11021921,19688757,35126020,62590629,111398910,198044551,351700332,623918086,1105715149
%N A261050 Expansion of Product_{k>=1} (1+x^k)^(Fibonacci(k)).
%H A261050 Vaclav Kotesovec, <a href="/A261050/b261050.txt">Table of n, a(n) for n = 0..1000</a>
%H A261050 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
%F A261050 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} (-1)^(k+1) * phi^k / ((phi^(2*k) - phi^k - 1)*k) = -0.3237251774053525012502809827680337358578568068831886835557918847... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
%F A261050 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k - x^(2*k)))). - _Ilya Gutkovskiy_, May 29 2018
%p A261050 f:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
%p A261050 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261050        add(binomial(f(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A261050     end:
%p A261050 a:= n-> b(n$2):
%p A261050 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 08 2015
%t A261050 nmax=40; CoefficientList[Series[Product[(1+x^k)^Fibonacci[k],{k,1,nmax}],{x,0,nmax}],x]
%Y A261050 Cf. A000045, A166861, A261051, A026007, A027998, A248882, A102866, A256142, A260916.
%K A261050 nonn
%O A261050 0,4
%A A261050 _Vaclav Kotesovec_, Aug 08 2015