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a(n) ~ n! / (2*(2+log(2)) * (log(1+log(2)/2))^(n+1)). - Vaclav Kotesovec, Jul 27 2021
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(PARI) my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021
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E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).