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%I #14 Nov 12 2023 12:46:24
%S 1,1,2,6,26,164,1540,22068,492616,17378968,977896328,88256247312,
%T 12819022165520,3002745820555664,1135759674922075168,
%U 694219521332053782624,686053892556368634929824,1096476587053610841771551296,2834651494015025836540377942080
%N Expansion of 1/(1 - x/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 5*x/(1 - 8*x/(1 - ... - Fibonacci(k)*x/(1 - ...)))))))), a continued fraction.
%H Alois P. Heinz, <a href="/A307082/b307082.txt">Table of n, a(n) for n = 0..99</a>
%F a(n) ~ c * phi^(n*(n+1)/2) / 5^(n/2), where phi = A001622 is the golden ratio and c = 10.15498753508843821457456033641336796744756370048241257586748102558791... - _Vaclav Kotesovec_, Sep 18 2021
%p b:= proc(x, y) option remember; (F->
%p `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-x+1), 0)+
%p `if`(y>x, b(x, y-1), 0)))(combinat[fibonacci])
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..18); # _Alois P. Heinz_, Nov 12 2023
%t nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-Fibonacci[k] x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
%Y Cf. A000045, A001622, A206741, A307083.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 23 2019
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