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%I #6 May 11 2022 10:37:21
%S 1,-2,-2,-16,-24,64,-720,-23808,-35840,282368,-3628800,-75458560,
%T -479001600,5315078144,-82614884352,-8601835798528,-20922789888000,
%U 321288633450496,-6402373705728000,-309168395474436096,-2379913632645120000,46441359567137275904
%N Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + tanh(x).
%t nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tanh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
%Y Cf. A155585, A353779, A353873, A353910, A353911.
%K sign
%O 1,2
%A _Ilya Gutkovskiy_, May 10 2022
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