Rational quadratic-coefficient conjecture for the limiting functions

Normalize the limiting functions $f$ and $\widehat f$ so that $f(0)=\widehat f(0)=1$, and consider their Taylor expansions at $0$. Rational quadratic-coefficient conjecture. For $n=8$, the quadratic coefficients of $f$ and $\widehat f$ are $-27/10$ and $-3/2$, respectively; for $n=24$, they are $-14347/5460$ and $-205/156$, respectively. The source presents these exact rational values as numerical conjectures; higher Taylor coefficients remain unexplained.