# integration – Reference to calculate a given integral with sine

Recently, I happened to discover a simple, elementary derivation of the following identity, valid for each one $$n, k in mathbb N$$:
begin {align *} frac1 pi int_0 ^ pi { rm d} x left (! frac { sin nx} { sin x} ! right) ^ {! ! k} = sum_ {i = 0} ^ k (-1) ^ i binom {k} {i} binom {k (n {+} 1) / 2-1-ni} {k-1} . day 1)} end {align *}
$$binom from$$ are the binomial coefficients, with the somewhat unusual convention that $$binom ab = 0$$ whenever $$a or $$a notin mathbb N_0$$which implies that, for example, with $$i geq k / 2$$ in the sum on the right are always zero (i.e. $$k$$ as upper limit could be replaced by $$lfloor k / 2 rfloor$$) and for yourself $$n$$ and odd $$k$$ the whole expression $$(1)$$ becomes zero.

I was looking for Integral $$(1)$$ in Gradshteyn & Ryzhik but only found the case $$k = 1$$ (see 3.612-2 on p. 391).
I also tried Mathematica, which has succeeded in the integral $$(1)$$ for specific $$n, k in mathbb N$$but, for example, the cases could not do that $$k = 1$$ or $$k = 2$$ for general $$n in mathbb N$$while after $$(1)$$ we just have
begin {align *} & frac1 pi int_0 ^ pi { rm d} x , frac { sin nx} { sin x} = begin {cases} 0 & n text {even} \ 1 & n text {odd} end {cases} , \ & frac1 pi int_0 ^ pi { rm d} x left (! frac { sin nx} { sin x} ! right) ^ {! ! 2} = n . day 2)} end {align *}
Can someone point out a reference where the integral lies $$(1)$$ or something similar is calculated?