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Expansion of Product_{k>=1} 1 / (1 + x^k)^Fibonacci(k).
2

%I #15 Apr 30 2023 15:45:04

%S 1,-1,0,-2,0,-3,0,-4,2,-5,8,0,26,19,74,74,195,221,464,560,1042,1258,

%T 2154,2536,3997,4341,6152,5204,5447,-1617,-10790,-39710,-83915,

%U -181639,-336564,-633844,-1108334,-1952371,-3293590,-5568202,-9148916,-15017471,-24144556,-38697396,-61005748,-95708150

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

%C Convolution inverse of A261050.

%F a(n) = Sum_{k=0..n} (-1)^k * A337009(n,k). - _Alois P. Heinz_, Apr 30 2023

%t nmax = 45; CoefficientList[Series[Product[1/(1 + x^k)^Fibonacci[k], {k, 1, nmax}], {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d) d Fibonacci[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 45}]

%Y Cf. A000045, A166861, A261050, A337009, A357179.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Oct 02 2022