Maclaurin series of $1-\left(\frac{\sin x}{x}\right) ^{\frac{2}{5}}$ has all coefficients positive I had a similar problem posted here, and after  experimentation with WA I noticed that the function $1-\left(\frac{\sin x}{x}\right) ^{\frac{2}{5}}$ has a Maclaurin expansion with all coefficients positive.
It seems to hold. A possible approach might using the product expansion for the function $\sin x$. Or maybe a differential equation.
Thank you for your interest!
$\bf{Added:}$ Denote the expression by $y(x)$. It's enough to show that $z\colon=y'$ has a positive expansion.  We have
$$x\cdot  z' =  \left(\frac{x^2}{1 - x \cot x} + \frac{3}{5}( 1- x \cot x) - 2 \right)\cdot z$$
If the expression in the brackets has a positive expansion (it seems so) then we can show that $z$ has a positive expasion.

Using the same approach as earlier
$$1-\left(\frac{\sin (x)}{x}\right)^a= \frac{a x^2}{6}\Bigg[1+\frac 1{60} \sum_{n=1}^\infty (-1)^n\frac{P_n(a)}{b_n }x^{2n}\Bigg]$$ where the first $b_n$ are
$$\{1,126,15120,997920,16345929600,\cdots\}$$ and the first $P_n(a)$ are
$$\left(
\begin{array}{cc}
n & P_n(a) \\
 1 & 5 a-2 \\
 2 & 35 a^2-42 a+16 \\
 3 & (5 a-4) \left(35 a^2-56 a+36\right) \\
 4 & 385 a^4-1540 a^3+2684 a^2-2288 a+768 \\
 5 & 175175 a^5-1051050 a^4+2862860 a^3-4252248 a^2+3327584 a-1061376
\end{array}
\right)$$
and all coefficients are positive as long as $a \leq \frac 25$
