The Chebyshev polynomial $U_n(x)$ of the second kind is characterized by 
$$
U_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}.
$$ 
It seems that
$$\operatorname*{Res}_x \left( U_n(x)+tU_{n-1}(x),\sum_{k=0}^{n-1}U_k(x) \right) =(-1)^{\frac{n(n-1)}{2}} t^{\left\lfloor\frac{k}{2} \right\rfloor}2^{n(n-1)},$$
where Res denotes the resultant of two polynomials. 

I do not know how to prove this "equality". Is it a known result? 
Similar results appeared in Dilcher and Stolarsky [1] theorem 2 and in Jacobs, Rayes and Trevisan [2].

**References**

[1] Karl Dilcher and Kenneth Stolarsky, "[Resultants and discriminants of Chebyshev and related polynomials](https://doi.org/10.1090/S0002-9947-04-03687-6)", Transactions of the American Mathematical Society, 357, pp. 965-981 (2004), [MR2110427](https://www.ams.org/mathscinet-getitem?mr=2110427), [Zbl 1067.12001](https://zbmath.org/?q=an%3A1067.12001). 

[2] David P. Jacobs and Mohamed O. Rayes and Vilmar Trevisan, "
[The resultant of Chebyshev polynomials](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D3FA1C0EBDAB83534C86375A0D711BF7/S0008439500017677a.pdf/the-resultant-of-chebyshev-polynomials.pdf)",
Canadian Mathematical Bulletin 54, No. 2, 288-296 (2011), [MR2884245](https://mathscinet.ams.org/mathscinet-getitem?mr=MR2884245), [Zbl 1272.12006](https://zbmath.org/?q=an%3A1272.12006).