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Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctan(x).
6

%I #6 May 10 2022 14:26:37

%S 1,0,-2,8,-16,96,-832,9344,-27648,238080,-4228608,55812096,-398991360,

%T 4930609152,-98606039040,2440552022016,-17762113880064,

%U 235149341884416,-7331825098948608,170578782435409920,-2009778629489197056,38563016760590598144,-1278044473427380666368

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

%t nn = 23; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcTan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

%Y Cf. A006973, A010050, A137852, A353611, A353818, A353819, A353821.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, May 08 2022