A354171 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + sin(x).

1, 0, -1, 4, -19, 44, -659, 8128, -18775, 67664, -3578279, 7629568, -476298835, 505198784, 25626362581, 4286437900288, -20903398375855, -118410655250176, -6399968826052559, -33100680116191232, 1010700510694925525, 706348515575880704, -1123931378903214542099

Offset: 1

Author

Ilya Gutkovskiy, May 18 2022

Keywords

• sign

Crossrefs

Cf. A067856, A170914, A170915, A328186, A328191, A353607, A353873, A354055, A354172, A354173, A354174, A354175, A354176.

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 + sin(x^k)) / k.