Prove the series has positive integer coefficients How can I show that the Maclaurin series for
$$
\mu(x) = (x^4+12x^3+14x^2-12x+1)^{-1/4}
\\
= 1+3\,x+19\,{x}^{2}+147\,{x}^{3}+1251\,{x}^{4}+11193\,{x}^{5}+103279\,
{x}^{6}+973167\,{x}^{7}+9311071\,{x}^{8}+\cdots
$$
has positive integer coefficients?  (I have others to do, too, but this one will be a start.)
possibilities
(a) The coefficients $Q(n)$ satisfy the recurrence
$$
(n+1)Q(n)+(12n+21)Q(n+1)+(14n+35)Q(n+2)+(-12n-39)Q(n+3)+(n+4)Q(n+4) = 0
$$
(b) $\mu(x)$ satisfies the differential equation
$$
(x^3+9x^2+7x-3)\mu(x)+(x^4+12x^3+14x^2-12x+1)\mu'(x)=0
$$
(c) factorization of $x^4+12x^3+14x^2-12x+1$ is
$$
\left( x-\sqrt {5}+3+\sqrt {15-6\,\sqrt {5}} \right)  \left( x-\sqrt 
{5}+3-\sqrt {15-6\,\sqrt {5}} \right)  \left( x+\sqrt {5}+3-\sqrt {15+
6\,\sqrt {5}} \right)  \left( x+\sqrt {5}+3+\sqrt {15+6\,\sqrt {5}}
 \right) 
$$
(d) $(1-8X)^{-1/4}$ has positive integer coefficients, But if $X$ is defined by $1-8X=x^4+12x^3+14x^2-12x+1$, then $X$ does not have integer coefficients.
(e) Can we compute the series for $\log\mu(x)$
$$3\,x+{\frac {29}{2}}{x}^{2}+99\,{x}^{3}+{\frac {3121}{4}}{x}^{4}+{
\frac {32943}{5}}{x}^{5}+{\frac {348029}{6}}{x}^{6}+\dots
$$
and then recognize that its exponential has integer coefficients?

I can prove the integer property, but not (yet) positivity.
Lemma: The coefficients of Taylor series of $\mu(x)$ are integers.
Proof. We can write
$$x^4+12x^3+14x^2-12x+1=\left(1-x\right)^4\Bigl(1-8\nu(x)\Bigr),$$
where $\nu(x)$ is given by
$$\nu(x)=\frac{2}{1-x}-\frac{7}{\left(1-x\right)^2}+\frac{7}{\left(1-x\right)^3}-\frac{2}{\left(1-x\right)^4}.$$
The coefficients of Taylor series of $\nu(x)$ are clearly integer and we have $\nu(0)=0$. Since the coefficients of $(1-x)^{-1}$ and $(1-8x)^{-\frac14}$ are integers, the statement follows. $\qquad\square$
Another option is to rewrite $\mu(x)$ in terms of $y=\frac{x}{1-x}$:
$$\mu(x)\mapsto \frac{1+y}{\left[1-8y\left(1-y^2\right)\left(1+2y\right)\right]^{\frac14}}.$$
It is interesting to note that even the coefficients of different powers of $y$ seem to be positive.
