The generating-function conjecture for $G(t,-k)$

Let $G(t,-k)$ denote the generating function considered in the paper, with $t$ a formal variable and $k$ an integer. Generating-function conjecture. For every integer $k\geq 1$,

$$G(t,-k)=\frac{2(k-1)!\,k!}{(2k)!}\left(\frac{\sin(t/2)}{t/2}\right)^{2k}\left[2^{2(k-2)+1}+\sum_{n=1}^{k-1}\left(\sum_{i=0}^{k-n-1}\binom{2k-1}{i}\right)\cos(nt)\right].$$

The formula is based on explicit computations through $k\leq 60$ and is presented as a conjectural closed form for these generating functions; the source gives no resolution status.