A354275 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctan(x).

1, 0, -2, 8, -16, -64, -832, 13824, 8192, -36096, -4228608, -58438656, -398991360, -3452915712, 44581613568, 7144463302656, -17762113880064, 126440605483008, -7331825098948608, -88237584523984896, 3154526750647517184, -27279757707305287680, -1278044473427380666368

Offset: 1

Author

Ilya Gutkovskiy, May 22 2022

Keywords

• sign

Crossrefs

Cf. A010050, A067856, A353820, A353915, A353972, A354117, A354175, A354274, A354276.

Programs

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = {1, 0, -1, 0}[[Mod[n, 4, 1]]]/n - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arctan(x^k)) / k.