A308446 Expansion of Product_{k>=1} 1/(1 - x^k)^Fibonacci(2*k).
1, 1, 4, 12, 39, 118, 371, 1129, 3468, 10524, 31910, 96155, 289016, 865000, 2581577, 7679762, 22784896, 67418329, 199004329, 586052299, 1722165404, 5050349249, 14781877481, 43185726143, 125949155473, 366716549379, 1066057177765, 3094398005409, 8969054893842
Offset: 0
Keywords
Programs
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Mathematica
nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^Fibonacci[2 k], {k, 1, nmax}], {x, 0, nmax}], x] 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}]
Formula
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
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