how to calculate the following integral related to Chebyshev polynomials Chebyshev polynomials of the second kind $V_n(x)$ can be defined as
$$V_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$
or through the recurrence relation
$$V_{n+1}=xV_n-V_{n-1}, V_0=1,  V_1=x.$$
First few low-order Chebyshev polynomials of the second
kind are as follows:
$$V_0=1, V_1=x,V_2=x^2-1,V_3=x^3-2x,$$
$$V_4=x^4-3x^2+1,V_5=x^5-4x^3+3x.$$
I want to know how to calculate the following integral relate to Chebyshev polynomials:
$$\int_0^\pi (\frac{\sin nx}{\sin x})^m dx$$
where $n,m\in \mathbb{Z}^+$.
It is easy to see the following result:
For an even number $n  \in \mathbb{Z}^+$ and and odd number $m \in \mathbb{N}$, we have
$$ \int_0^\pi (\frac {\sin nx}{\sin x})^{m} dx=0.$$
I conjectured   that the result is a polynomial $P(n)$ of order $m−1$. But I have no idea about the proof. 
I prefer to know the other two cases beside the above special case.   Thank you.
 A: We use complex analysis, of course, standard way to convert an integral of a periodic function, integrated on a period.
In our case:
$$
\begin{aligned}
J(n,m)
&=
\int_0^\pi\left(\frac {\sin nt}{\sin t}\right)^m\; dt
\\
&=
\frac 12
\int_0^{2\pi}
\left(\frac {e^{int}-e^{-int}}{e^{it}-e^{-it}}\right)^m
\cdot\frac 1{e^{it}}\; ie^{it}\; dt
\\
&=
\frac 1{2i}
\int_\gamma
\left(\frac {z^{n}-z^{-n}}{z-z^{-1}}\right)^m\cdot\frac 1z\; 
\; dz
\\
&=
2\pi i\cdot\frac 1{2i}
\cdot\text{Residue of }
\frac 1z
\underbrace{\left(z^{n-1}+\dots z^{n-3}+\dots z^{-(n-3)}+z^{-(n-1)}\right)^m}_{P(n,m)}\ .
\end{aligned}
$$
Now we have to isolate the residue of the above integral on $\gamma$, the unit circle. We can simplify and need the coefficient of degree zero in 
$$
P(n,m)=\left(z^{n-1}+\dots z^{n-3}+\dots z^{-(n-3)}+z^{-(n-1)}\right)^m\ .
$$
From here, as in the answer of Carlo Beenakker pointing to an OEIS.
Computer check in a special case.
sage: integral( ( sin(7*x) / sin(x) )^9, x, 0, pi )
2636263*pi
sage: var('z');
sage: P = (( z^6 + z^4 + z^2 + 1 + z^-2 + z^-4 + z^-6 )^9) / z
sage: P.residue(z)
2636263
sage: # also

