A353972 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsin(x).

1, 0, 1, -4, 29, -124, 1583, -17088, 124553, -1152816, 20127867, -262838016, 3978820221, -48595514304, 914656587063, -24441484099584, 370244721585681, -5884988565575424, 162968423791332339, -3855257807841017856, 82014901819948738629, -1934570487417807744000, 58311771938510122952559

Offset: 1

Author

Ilya Gutkovskiy, May 22 2022

Keywords

• sign

Crossrefs

Cf. A001818, A067856, A353818, A353913, A354115, A354171, A354274, A354275, 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] = Mod[n, 2] (n - 2)!!/(n (n - 1)!!) - 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 + arcsin(x^k)) / k.