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%I #6 May 23 2022 09:12:30
%S 1,0,-2,8,-16,-64,-832,13824,8192,-36096,-4228608,-58438656,
%T -398991360,-3452915712,44581613568,7144463302656,-17762113880064,
%U 126440605483008,-7331825098948608,-88237584523984896,3154526750647517184,-27279757707305287680,-1278044473427380666368
%N Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctan(x).
%F E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arctan(x^k)) / k.
%t 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}]
%Y Cf. A010050, A067856, A353820, A353915, A353972, A354117, A354175, A354274, A354276.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, May 22 2022
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