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%I #6 May 10 2022 14:26:41
%S 1,0,2,-8,64,-384,3968,-34432,414720,-4454400,68247552,-912236544,
%T 15949529088,-245572583424,5012834549760,-92436465352704,
%U 2119956936523776,-42836227522560000,1123874181449515008,-26161653829651660800,730049769522063212544,-18719979459270521389056
%N Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctanh(x).
%t nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcTanh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
%Y Cf. A006973, A010050, A137852, A353779, A353818, A353819, A353820.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, May 08 2022
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