= Coefficient sign lemma for the first Dahlquist barrier
{title2=$x/\operatorname{arctanh}x=1-\sum_{j\geq1}\gamma_jx^{2j},\quad\gamma_j>0$}
Write $F(x)=\int_0^1(1+x)^t(1-x)^{1-t}\,dt=x/\operatorname{arctanh}x$. Pairing $t$ and $1-t$ after differentiating twice gives
$$
F''(x)=-4\int_0^1t(1-t)(1-x^2)^{-3/2}\cosh[(2t-1)\operatorname{arctanh}x]\,dt.
$$
The <power series> of $(1-x^2)^{-3/2}$ has strictly positive even coefficients, and the other factor has nonnegative even coefficients. Thus every even coefficient of $F''$ is negative. Since $F$ is even with $F(0)=1$, the displayed sign pattern follows. This supplies a short positivity argument for the <first Dahlquist barrier>.
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