A308446 - OEIS

%I #8 May 28 2019 09:48:57

%S 1,1,4,12,39,118,371,1129,3468,10524,31910,96155,289016,865000,

%T 2581577,7679762,22784896,67418329,199004329,586052299,1722165404,

%U 5050349249,14781877481,43185726143,125949155473,366716549379,1066057177765,3094398005409,8969054893842

%N Expansion of Product_{k>=1} 1/(1 - x^k)^Fibonacci(2*k).

%C Euler transform of A001906.

%F a(n) ~ phi^(2*n) * exp(2*sqrt(n)/5^(1/4) - 3/10 + S) / (2 * 5^(1/8) * sqrt(Pi) * n^(3/4)), where S = Sum_{k>=2} 1/((phi^(2*k) - 3 + 1/phi^(2*k))*k) = 0.155349347463140787939176213528043741704916536093946010733676987281... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, May 28 2019

%t nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^Fibonacci[2 k], {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Fibonacci[2 d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

%Y Cf. A000045, A001906, A032170, A034691, A088305, A166861.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 27 2019