A357475 Expansion of Product_{k>=1} 1 / (1 + x^k)^Fibonacci(k).

1, -1, 0, -2, 0, -3, 0, -4, 2, -5, 8, 0, 26, 19, 74, 74, 195, 221, 464, 560, 1042, 1258, 2154, 2536, 3997, 4341, 6152, 5204, 5447, -1617, -10790, -39710, -83915, -181639, -336564, -633844, -1108334, -1952371, -3293590, -5568202, -9148916, -15017471, -24144556, -38697396, -61005748, -95708150

Offset: 0

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

• sign

Comments

Convolution inverse of A261050.

Crossrefs

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

Programs

Mathematica

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

Formula

a(n) = Sum_{k=0..n} (-1)^k * A337009(n,k). - Alois P. Heinz, Apr 30 2023